#7624738

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319518, #72320055, #72320368
maximal normal subgroup of	#72319518, #72320055, #72320368

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531934);