#7624940

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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

1227 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)(20,29)(21,32)(23,34)(25,35)(26,38)(28,39)(30,46)(31,42)(33,45)(36,48)(37,43)(40,41)(44,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18)(19,28)(20,25)(21,26)(22,24)(23,31)(27,39)(29,35)(30,44)(32,38)(33,37)(34,42)(36,40)(41,48)(43,45)(46,47);(17,19,29,41,21,30,33,42,22,31,45,46,32,40,20,27)(18,23,35,47,26,36,37,39,24,28,43,48,38,44,25,34);(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)(23,28)(25,37)(26,38)(34,39)(35,43)(36,44)(47,48)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=f,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287059, #72320187, #72320269
maximal normal subgroup of	#72287059, #72320187, #72320269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532136);