#405576

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#23443	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1396 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)(11,21)(12,26)(13,20)(14,18)(15,19)(16,23)(17,29)(22,31)(24,32)(25,30)(27,34)(28,33)(35,38)(36,40)(37,39);(2,5)(6,8)(9,11,14,25,15,24,13,23)(10,16,20,32,19,30,18,21)(12,22,29,38,27,36,28,37)(17,31,26,39,33,40,34,35);(12,27)(16,30)(17,33)(21,32)(22,36)(26,34)(28,29)(37,38);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,27,14,29)(10,17,18,34,19,33,20,26)(11,22,23,37,24,36,25,38)(16,31,21,35,30,40,32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312772);