#6684824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2007 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=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, 6592020);