#358520

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1174 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)(3,6)(4,5)(7,8)(9,10,14,18,15,20,13,19)(11,16,23,31,24,32,22,35)(12,25,29,34,27,36,28,17)(21,37,39,40,38,33,26,30);(2,5)(6,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,33)(18,31)(19,35)(20,32)(21,29)(25,30)(27,39)(28,38)(34,37)(36,40);(9,12,13,28,15,27,14,29)(10,17,18,25,20,34,19,36)(11,21,22,26,24,38,23,39)(16,30,31,37,32,40,35,33);(1,3,4,7)(2,6,5,8)(12,27)(17,34)(21,38)(25,36)(26,39)(28,29)(30,40)(33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265716);