#7624340

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531536);