#363963

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2564 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=i,h^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, 271159);