#6545004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452200);