#7624932

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
prime radical	#33733	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1199 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)(20,28)(21,31)(23,33)(25,34)(26,38)(29,43)(30,41)(32,42)(35,44)(36,48)(37,47)(39,40)(45,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,22,24)(19,23,30,35)(20,25,32,37)(21,26,31,38)(27,33,41,44)(28,34,42,47)(29,36,39,45)(40,46,43,48);(17,19,28,40,21,29,32,41,22,30,42,43,31,39,20,27)(18,23,34,46,26,36,37,44,24,35,47,48,38,45,25,33);(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)(23,35)(25,37)(26,38)(33,44)(34,47)(36,45)(46,48)

Polycyclic presentation:

2^9|b^2=fi,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,d^b=df,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532128);