#6389200

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60453	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,d^c=di,e^d=ei

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, 6296396);