#6585789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574008
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263842, #72263847, #72263865
maximal normal subgroup of	#72263842, #72263847, #72263865

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492985);