#6627371

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#471
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#41	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534567);