#414921

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
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

1010 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)(17,27)(22,32)(24,35)(26,34)(28,33)(29,31)(30,38)(36,37)(39,40);(9,11,13,23,15,24,14,29)(10,16,18,21,20,31,19,35)(12,22,26,37,28,38,27,39)(17,32,33,40,34,30,25,36);(9,12,13,26,15,28,14,27)(10,17,18,33,20,34,19,25)(11,22,23,37,24,38,29,39)(16,30,21,36,31,32,35,40);(1,3,4,8)(2,5,6,7)

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

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, 322117);