#497100

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

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

normal subgroups only

This group has 474 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) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,32)(24,28)(25,39)(26,31)(27,35)(29,37)(30,38)(33,45)(34,46)(36,48)(40,41)(42,50)(43,49)(44,52)(47,51);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,22)(23,27)(24,30)(25,29)(26,31)(28,38)(32,35)(33,41)(34,43)(36,44)(37,39)(40,45)(42,51)(46,49)(47,50)(48,52);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,23,33,49,24,34,25,35,26,36,37,50,38,51,40,52)(22,27,41,46,30,43,29,32,31,44,39,47,28,42,45,48);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,24,26,38)(22,28,31,30)(23,34,36,51)(25,37,40,33)(27,42,44,43)(29,41,45,39)(32,46,48,47)(35,50,52,49)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efhi,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, 404296);