#6520449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2473 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)(14,20)(15,25)(17,18)(22,26)(23,24)(27,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20,19,23,18,24)(15,16,22,27,26,21,25,28);(6,9)(7,11)(13,15,17,22,19,26,18,25)(14,21,20,28,23,16,24,27);(5,7,8,11)(6,10,9,12)(13,16,19,21)(14,22,23,25)(15,20,26,24)(17,27,18,28);(6,9)(10,12)(14,23)(16,21)(20,24)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427645);