#6671475

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2258 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578671);