#502856

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

839 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410052);