#236578

order	512
nilpotency class	3
isoclinism	isoclinic class of #236557
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	#60967	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1139 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,35)(28,43)(29,38)(30,41)(31,39)(33,40)(34,42)(37,46)(44,48)(45,47);(1,3)(2,8)(4,12)(5,9)(6,14)(7,15)(10,13)(11,16)(17,19,23,31)(18,24,27,39)(20,28,34,45)(21,30,22,29)(25,37,40,48)(26,41,36,38)(32,47,42,43)(33,46,35,44);(1,4)(2,9)(3,11)(5,16)(6,13)(7,8)(10,15)(12,14)(17,20)(18,25)(19,28)(21,35)(22,33)(23,34)(24,37)(26,42)(27,40)(29,46)(30,44)(31,45)(32,36)(38,47)(39,48)(41,43);(1,5,7,6)(2,10,12,11)(3,9,15,14)(4,16,8,13)(17,21,23,22)(18,26,27,36)(19,29,31,30)(20,33,34,35)(24,38,39,41)(25,32,40,42)(28,44,45,46)(37,43,48,47)

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=di,e^c=ei,f^c=fi,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, 143774);