#239318

order	512
nilpotency class	3
isoclinism	isoclinic class of #239311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3186	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

873 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,28)(20,32)(21,26)(22,37)(23,27)(24,30)(25,35)(29,46)(31,40)(33,45)(34,44)(36,43)(38,41)(39,47)(42,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,42)(26,40)(27,41)(28,37)(29,35)(32,39)(34,48)(36,47)(43,46)(44,45);(6,12)(13,16)(17,20,22,34,23,35,21,36)(18,25,26,44,27,32,37,43)(19,29,30,48,31,33,38,47)(24,39,28,45,41,42,40,46);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,22)(18,26,27,37)(19,30,31,38)(20,34,35,36)(24,40,41,28)(25,43,32,44)(29,47,33,48)(39,45,42,46)

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,f^c=fh,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, 146514);