#414493

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

819 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)(3,8)(4,6)(5,7)(12,23)(25,36)(28,39)(33,40);(1,3,5,4)(2,6,7,8)(9,10)(11,19)(12,16)(13,26)(14,18)(15,21)(17,24)(20,33)(22,32)(23,30)(25,37)(27,29)(28,35)(31,36)(34,40)(38,39);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,28)(17,32)(18,29)(19,26)(21,24)(23,34)(25,38)(30,39)(31,40)(33,37)(35,36);(9,12,24,36,14,23,13,25)(10,16,17,31,18,30,26,37)(11,20,21,35,22,34,27,38)(15,28,32,40,29,39,19,33)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321689);