#363961

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271157);