#235303

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

1286 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)(3,7)(4,12)(5,9)(6,11)(8,14)(10,16)(13,15)(17,18)(19,23)(20,30)(21,33)(22,26)(24,32)(25,35)(27,42)(28,37)(29,46)(31,41)(34,40)(36,44)(38,45)(39,47)(43,48);(1,3)(2,6)(4,13)(5,10)(7,12)(8,16)(9,15)(11,14)(17,19)(18,23)(20,27)(21,29)(22,28)(24,39)(25,38)(26,37)(30,48)(31,43)(32,44)(33,46)(34,47)(35,45)(36,40)(41,42);(1,3)(2,7)(4,11)(5,10)(6,12)(8,15)(9,16)(13,14)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,41)(26,40)(28,43)(29,47)(30,33)(32,35)(37,39)(38,48)(42,46)(44,45);(3,10)(6,15)(7,16)(11,13)(19,28)(23,37)(27,43)(29,45)(36,39)(38,46)(42,48)(44,47)

Polycyclic presentation:

2^9|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, 142499);