#6382798

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2731 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=gi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289994);