#235634

order	512
nilpotency class	3
isoclinism	isoclinic class of #235627
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
prime radical	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1167 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142830);