#6816787

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2216 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)(5,6,9,11)(7,12,8,10)(13,14)(15,18)(16,27)(17,25)(19,24)(20,23)(21,26)(22,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,21)(16,17)(18,24)(19,26)(20,28)(22,25)(23,27);(1,2,3,4)(13,16,17,15,19,28,20,26)(14,22,25,21,24,27,23,18);(5,7,9,8)(6,10,11,12)(13,17,19,20)(14,23,24,25)(15,16,26,28)(18,22,21,27);(5,8,9,7)(6,10,11,12)(13,18,17,22,19,21,20,27)(14,15,25,28,24,26,23,16)

Polycyclic presentation:

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

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, 6723983);