#7623440

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
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
prime radical	#3058	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1228 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)(19,24)(20,26)(21,25)(22,36)(23,28)(27,34)(29,37)(30,38)(31,41)(32,39)(33,46)(35,40)(42,45)(43,47)(44,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,22)(18,24,28,36)(20,29,31,33)(21,30,35,34)(25,38,40,27)(26,37,41,46)(32,43,44,45)(39,47,48,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,22)(21,34)(24,27)(26,39)(28,40)(29,33)(30,35)(32,45)(36,38)(37,42)(41,48)(43,44)(46,47);(2,8)(6,9)(7,16)(14,15)(17,20)(18,26)(19,29)(21,32)(22,33)(23,31)(24,37)(25,39)(27,42)(28,41)(30,43)(34,45)(35,44)(36,46)(38,47)(40,48);(20,31)(26,41)(29,33)(32,44)(37,46)(39,48)(42,47)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530636);