#357721

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264917);