#388010

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1004 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=eg,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295206);