#7624693

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
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 ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1237 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=i,e^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=dfi,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319166, #72320269, #72320683
maximal normal subgroup of	#72319166, #72320269, #72320683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531889);