#382817

order	512
nilpotency class	3
isoclinism	isoclinic class of #213736
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1193 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,27)(20,29)(21,28)(22,35)(23,30)(24,31)(25,33)(26,32)(34,38)(36,40)(37,39);(9,11,14,21)(10,15,17,28)(12,22,23,19)(13,25,26,20)(16,27,30,35)(18,33,32,29)(24,34,37,36)(31,40,39,38);(2,5)(6,8)(9,12,13,24,14,23,26,37)(10,16,18,31,17,30,32,39)(11,19,20,34,21,22,25,36)(15,27,29,38,28,35,33,40);(1,3,4,7)(2,6,5,8)(12,23)(13,26)(16,30)(18,32)(19,22)(20,25)(27,35)(29,33)

Polycyclic presentation:

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

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, 290013);