#6473345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2550 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)(5,6)(7,9)(8,10)(11,12)(14,20)(15,21)(17,18)(22,23)(25,26)(27,28);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14,18,22,19,23,17,20)(15,24,21,28,26,16,25,27);(7,11)(9,12)(13,15,18,21,19,26,17,25)(14,16,22,27,23,24,20,28);(7,11)(9,12)(13,16)(14,21)(15,20)(17,28)(18,27)(19,24)(22,26)(23,25);(6,10)(9,12)(13,17,19,18)(14,20,23,22)(15,21,26,25)(16,27,24,28)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,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, 6380541);