#7624966

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^c=ei

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, 7532162);