#357960

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265156);