#6685563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308293
maximal normal subgroup of	#72308293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592759);