#7500829

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408025);