#414922

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

996 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,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, 322118);