#7500983

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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3176 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)(9,10,14,20)(11,21,24,16)(12,25,27,35)(13,19,15,30)(17,26,34,28)(18,23,32,29)(22,37,39,33)(31,38,36,40);(1,2)(3,4)(5,6)(7,8)(9,11,14,24)(10,16,20,21)(12,22,27,39)(13,29,15,23)(17,31,34,36)(18,19,32,30)(25,33,35,37)(26,40,28,38);(6,8)(9,12,14,27)(10,17,20,34)(11,22,24,39)(13,28,15,26)(16,31,21,36)(18,33,32,37)(19,35,30,25)(23,40,29,38);(2,4)(9,11)(10,18)(12,22)(13,23)(14,24)(15,29)(16,30)(17,33)(19,21)(20,32)(25,31)(26,38)(27,39)(28,40)(34,37)(35,36);(9,13,14,15)(10,19,20,30)(11,23,24,29)(12,26,27,28)(16,18,21,32)(17,25,34,35)(22,38,39,40)(31,37,36,33)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^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,e^c=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, 7408179);