#6811386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

2146 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)(5,6)(7,10)(8,12)(9,11)(13,14)(15,24)(16,27)(17,28)(18,26)(19,23)(20,22)(21,25);(5,7,10,6)(8,12,9,11)(13,15)(14,18)(16,20)(17,25)(19,26)(21,22)(23,24)(27,28);(2,4)(13,16,17,26,19,25,20,15)(14,21,28,24,23,27,22,18);(5,8,10,9)(6,11,7,12)(13,17,19,20)(14,22,23,28)(15,25,26,16)(18,21,24,27);(5,9,10,8)(6,11,7,12)(13,18,20,27,19,24,17,21)(14,15,22,25,23,26,28,16)

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=f,e^2=fhi,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

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, 6718582);