#6440272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

2367 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347468);