#497134

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 76 points:

(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14)(15,25)(16,30)(17,22)(18,38)(19,40)(20,27)(21,34)(23,39)(24,37)(26,51)(28,57)(29,43)(31,62)(32,48)(33,59)(35,69)(36,45)(41,63)(42,52)(44,53)(46,66)(47,65)(49,58)(50,56)(54,76)(55,70)(60,68)(61,67)(64,73)(71,75)(72,74);(1,3,4,2)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,31)(17,27)(18,28)(19,29)(21,44)(22,25)(23,42)(24,43)(26,33)(30,51)(32,52)(34,63)(35,64)(36,55)(37,56)(38,70)(39,58)(40,50)(41,46)(45,57)(47,72)(48,49)(53,66)(54,67)(59,62)(60,71)(61,73)(65,75)(68,74)(69,76);(5,8,10,9)(6,11,7,12)(13,16,32,65,19,35,18,34,17,33,39,68,37,67,36,66)(14,21,45,69,24,47,23,30,22,46,38,61,40,60,48,59)(15,26,52,74,29,54,28,53,27,31,58,75,56,64,55,63)(20,41,57,73,43,71,42,62,25,44,70,76,50,72,49,51);(5,9,10,8)(6,11,7,12)(14,22)(15,27)(21,46)(23,48)(24,40)(26,31)(28,55)(29,56)(30,59)(38,45)(47,60)(52,58)(53,63)(54,64)(61,69)(74,75)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404330);