#6473416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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

2399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=i,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,e^d=ei

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, 6380612);