#6558272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2713 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)(15,23,26,27)(16,20,28,25)(17,21,18,24);(5,6)(7,10)(8,9)(11,12)(13,15,17,20,19,26,18,25)(14,16,24,27,22,28,21,23);(1,2,3,4)(6,9)(10,12)(13,16,19,28)(14,20,22,25)(15,24,26,21)(17,27,18,23);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,21,22,24)(15,20,26,25)(16,23,28,27);(7,11)(10,12)(13,17,19,18)(14,21,22,24)(15,25,26,20)(16,27,28,23)

Polycyclic presentation:

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

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, 6465468);