#6470735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2230 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377931);