#6382770

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2818 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289966);