#6583729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263916, #72263954, #72263982
maximal normal subgroup of	#72263916, #72263954, #72263982

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490925);