#232668

order	512
nilpotency class	3
isoclinism	isoclinic class of #232654
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139864);