#405697

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1115 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)(9,10,14,18)(11,15,20,29)(12,16,24,33)(13,25,27,17)(19,35,36,28)(21,38,37,30)(22,31,39,40)(23,34,26,32);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,22,24,39)(13,21)(14,20)(16,31,33,40)(17,30)(18,29)(19,23,36,26)(25,38)(27,37)(28,32,35,34);(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,34)(18,33)(20,36)(21,39)(22,37)(23,27)(25,32)(29,35)(30,40)(31,38);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,30)(18,29)(22,23)(24,36)(25,38)(26,39)(27,37)(31,32)(33,35)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi,f^d=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, 312893);