#7500956

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#35634	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7408152);