#6816851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

2302 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)(15,24)(17,26)(20,27)(22,25);(5,6)(7,10)(8,12)(9,11)(13,14)(15,17)(16,21)(18,23)(19,28)(20,25)(22,27)(24,26);(1,2,3,4)(13,15,16,22,18,24,19,25)(14,20,28,26,23,27,21,17);(5,7,9,8)(6,10,11,12)(13,16,18,19)(14,21,23,28)(15,25,24,22)(17,20,26,27);(5,8,9,7)(6,10,11,12)(13,17,19,27,18,26,16,20)(14,22,21,15,23,25,28,24)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=fh,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,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724047);