#378899

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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:

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

Polycyclic presentation:

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

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, 286095);