#6817200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei

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, 6724396);