#6382776

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2742 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)(9,10,14,18,15,20,13,19)(11,21,23,16,25,31,24,32)(12,17,29,35,27,34,28,36)(22,33,38,30,37,40,26,39);(5,7,8,6)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,33)(18,21)(19,31)(20,32)(22,28)(27,38)(29,37)(30,36)(34,40)(35,39);(1,2,3,4)(12,27)(17,34)(22,37)(26,38)(28,29)(30,39)(33,40)(35,36);(9,12,13,28,15,27,14,29)(10,17,19,36,20,34,18,35)(11,22,24,26,25,37,23,38)(16,30,21,33,32,39,31,40);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,27,29)(16,21,32,31)(17,35,34,36)(22,38,37,26)(30,33,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,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,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289972);