#6585505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1989 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285049, #72285079, #72285305
maximal normal subgroup of	#72285049, #72285079, #72285305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492701);