#381138

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

1133 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,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, 288334);