#6502713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500120
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

2215 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)(13,14,18,21,17,22,16,26)(15,23,25,28,24,19,27,20);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,24)(14,19,22,23)(16,27,18,25)(20,26,28,21);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,24)(18,27)(19,21)(22,28)(23,26);(7,12)(9,11)(15,24)(19,23)(20,28)(25,27);(6,10)(9,11)(13,16,17,18)(14,21,22,26)(15,25,24,27)(19,28,23,20)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6409909);