#236696

order	512
nilpotency class	3
isoclinism	isoclinic class of #236695
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	#60805	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1197 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:

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

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,e^c=ei,f^c=fhi,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, 143892);