#6567604

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6,10,9)(7,12,8,11)(13,14)(15,24)(16,26)(17,22)(18,28)(19,23)(20,21)(25,27);(5,7,10,8)(6,11,9,12)(13,15,18,21,19,26,17,25)(14,20,22,24,23,27,28,16);(2,4)(5,6,10,9)(7,12,8,11)(13,16,18,20,19,24,17,27)(14,21,22,26,23,25,28,15);(5,8,10,7)(6,11,9,12)(13,17,19,18)(14,22,23,28)(15,25,26,21)(16,20,24,27);(5,9,10,6)(7,12,8,11)(13,17,19,18)(14,22,23,28)(15,21,26,25)(16,27,24,20)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474800);