#331943

order	512
nilpotency class	3
isoclinism	isoclinic class of #331927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18857	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18,22,27)(19,28,32,42)(20,24,35,41)(21,25,36,26)(23,30,38,31)(29,44,33,43)(34,39,47,40)(37,46,48,45);(1,2)(3,8)(4,9)(5,7)(6,10)(11,12)(13,16)(14,15)(17,19)(18,23)(20,33)(21,30)(22,32)(24,37)(25,28)(26,42)(27,38)(29,35)(31,36)(34,46)(39,43)(40,44)(41,48)(45,47);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,20,22,35)(18,24,27,41)(19,29,32,33)(21,34,36,47)(23,37,38,48)(25,39,26,40)(28,43,42,44)(30,45,31,46);(1,3,6,13)(2,8,10,16)(4,14,5,12)(7,11,9,15)(18,25)(19,30)(21,36)(24,39)(26,27)(28,42)(29,45)(31,32)(33,46)(34,47)(40,41)(43,44)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239139);