#357829

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

2690 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:

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

Polycyclic presentation:

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

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, 265025);