#235313

order	512
nilpotency class	3
isoclinism	isoclinic class of #235303
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

1290 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)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18)(19,29)(20,33)(21,27)(22,26)(23,28)(24,31)(25,36)(30,38)(32,43)(34,42)(35,41)(37,39)(40,45)(44,46)(47,48);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,40)(26,43)(27,29)(28,39)(33,44)(34,45)(35,47)(36,46)(38,41)(42,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,15)(10,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,38)(26,33)(27,42)(28,41)(29,44)(31,47)(32,45)(37,46)(39,40)(43,48);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(20,34)(25,41)(30,45)(33,42)(35,36)(38,40)(44,48)(46,47)

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=dhi,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, 142509);