#6544795

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2122 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)(13,14,20,25,18,16,19,22)(15,23,26,27,17,21,24,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,23)(17,18)(19,24)(20,26)(22,27)(25,28);(6,9)(10,12)(13,16,18,14)(15,21,17,23)(19,22,20,25)(24,28,26,27);(5,7)(6,10)(8,11)(9,12)(13,17,18,15)(14,21,16,23)(19,26,20,24)(22,27,25,28);(7,11)(10,12)(15,17)(21,23)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,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, 6451991);