#6673251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,28,20)(16,29,31,38)(17,24,19,22)(21,33,36,30)(26,40,42,44)(27,35,32,37)(34,43,39,41);(1,2)(3,4)(6,9)(7,11)(13,15)(14,20)(16,26)(17,32)(18,28)(19,27)(21,34)(22,37)(23,25)(24,35)(29,44)(30,43)(31,42)(33,41)(36,39)(38,40);(2,4)(13,16,18,31)(14,21,23,36)(15,26,28,42)(17,30,19,33)(20,34,25,39)(22,29,24,38)(27,41,32,43)(35,40,37,44);(6,9)(10,12)(13,17,18,19)(14,22,23,24)(15,27,28,32)(16,30,31,33)(20,35,25,37)(21,29,36,38)(26,41,42,43)(34,40,39,44);(2,4)(5,7,8,11)(6,10,9,12)(15,28)(16,31)(20,25)(21,36)(27,32)(29,38)(30,33)(35,37)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=fi,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,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580447);