#405453

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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	#60444	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,24)(13,27)(14,17)(15,22)(16,25)(18,20)(21,32)(23,33)(26,31)(28,37)(29,34)(30,39)(35,38)(36,40);(1,3,4,7)(2,5,6,8)(9,11,20,34,14,23,13,22)(10,15,27,33,17,29,18,19)(12,21,35,39,25,36,26,37)(16,30,38,32,24,28,31,40);(9,12,13,26,14,25,20,35)(10,16,18,31,17,24,27,38)(11,21,22,37,23,36,34,39)(15,28,19,32,29,30,33,40);(2,6)(5,8)(10,17)(12,25)(15,29)(18,27)(19,33)(21,36)(26,35)(37,39)

Polycyclic presentation:

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

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, 312649);