#6672956

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

2285 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);(6,9)(7,11)(13,14)(15,19)(16,18)(17,21)(20,24)(22,27)(23,25)(26,28);(2,4)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(6,9)(10,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,26)(21,27)(24,28);(5,7,8,11)(6,10,9,12)(14,18)(15,22)(16,24)(19,26)(20,21)(25,27)

Polycyclic presentation:

2^9|e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,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	#72308783, #72308867
maximal normal subgroup of	#72308783, #72308867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580152);