#6669114

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2305 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308980, #72308983, #72308985
maximal normal subgroup of	#72308980, #72308983, #72308985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576310);