#6816850

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
prime radical	C₂×#3282	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,9,11)(7,12,8,10)(13,14,18,22)(15,19,24,32)(16,26,29,20)(17,21,30,36)(23,31,37,42)(25,35,28,33)(27,40,39,34)(38,44,41,43);(5,6)(7,10)(8,12)(9,11)(15,23)(16,27)(17,30)(19,31)(20,34)(21,36)(24,37)(25,28)(26,40)(29,39)(32,42)(33,35);(1,2,3,4)(13,15,17,23,18,24,30,37)(14,19,21,31,22,32,36,42)(16,28,27,41,29,25,39,38)(20,35,34,44,26,33,40,43);(5,7,9,8)(6,10,11,12)(15,24)(19,32)(23,37)(25,28)(31,42)(33,35)(38,41)(43,44);(5,8,9,7)(6,10,11,12)(13,16,17,27,18,29,30,39)(14,20,21,34,22,26,36,40)(15,25,23,38,24,28,37,41)(19,33,31,43,32,35,42,44)

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,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, 6724046);