#6407406

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314602);