#381127

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1137 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,f^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^c=fi,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, 288323);