#7622734

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3444	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1263 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,20,25,21,26,30,38,22,24,31,36,32,37,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,33)(25,34)(26,35)(30,41)(31,39)(32,40)(36,43)(37,44)(38,45)(42,47)(46,48);(3,10)(6,9)(11,16)(13,15)(18,24)(19,27)(25,36)(26,37)(28,39)(29,40)(38,46)(41,47);(2,7)(6,9)(8,14)(13,15)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Polycyclic presentation:

2^9|a^2=g,e^2=f,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cfi,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529930);