#6567598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2036 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474794);