#6502678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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

2193 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)(13,14,19,22,18,24,17,23)(15,16,25,28,21,20,26,27);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15,18,21)(14,20,24,16)(17,26,19,25)(22,27,23,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,24)(17,28)(18,20)(19,27)(22,25)(23,26);(7,12)(9,11)(13,17,18,19)(14,22,24,23)(15,25,21,26)(16,27,20,28);(6,10)(9,11)(13,17,18,19)(14,23,24,22)(15,25,21,26)(16,28,20,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6409874);