#235026

order	512
nilpotency class	3
isoclinism	isoclinic class of #235012
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1196 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:

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

Polycyclic presentation:

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

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, 142222);