#434692

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341888);