#235290

order	512
nilpotency class	3
isoclinism	isoclinic class of #235276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142486);