#6658557

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24162	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi,e^d=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, 6565753);