#374288

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281484);