#354613

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fi,d^2=gi,h^2=i,b^a=be,c^a=cf,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261809);