#361991

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1422 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=i,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,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, 269187);