#6393317

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,32,38)(19,23,29,35,27,33,40,44)(20,24,31,37,30,36,42,46)(28,34,41,45,39,43,47,48);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48);(10,14)(13,16)(17,19,21,29,22,27,32,40)(18,23,25,35,26,33,38,44)(20,28,31,41,30,39,42,47)(24,34,37,45,36,43,46,48);(1,4,6,5)(2,7,3,8)(17,20)(18,24)(19,28)(21,31)(22,30)(23,34)(25,37)(26,36)(27,39)(29,41)(32,42)(33,43)(35,45)(38,46)(40,47)(44,48);(1,5,6,4)(2,7,3,8)(20,30)(24,36)(28,39)(31,42)(34,43)(37,46)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,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, 6300513);