#235635

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	#60802	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1199 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:

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

Polycyclic presentation:

2^9|b^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, 142831);