#6660363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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	#37544	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(11,19)(12,16)(13,17)(14,18)(15,22)(20,24)(21,23)(25,26,30,33)(27,31,32,40)(28,37,38,36)(29,35,39,34);(1,2,4,5)(3,6,7,8)(9,11,14,22)(10,15,18,19)(12,20,13,21)(16,23,17,24)(25,27,30,32)(26,31,33,40)(28,34,38,35)(29,36,39,37);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24)(25,28,29,31,30,38,39,40)(26,32,34,37,33,27,35,36);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,19,24)(25,27)(26,31)(28,35)(29,36)(30,32)(33,40)(34,38)(37,39);(9,13,14,12)(10,16,18,17)(11,21,22,20)(15,23,19,24)(26,33)(27,32)(34,35)(36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fghi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,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, 6567559);