#6425103

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	C₂³	C₂⁴×C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=df,e^c=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, 6332299);