#332012

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
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	#18946	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

529 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)(17,18,23,28)(19,29,33,39)(20,34,36,43)(21,27,22,26)(24,31,40,32)(25,37,42,35)(30,41,47,38)(44,45,48,46);(1,2)(3,8)(4,9)(5,7)(6,10)(11,12)(13,16)(14,15)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,41)(26,39)(27,29)(28,40)(34,44)(35,45)(36,47)(37,46)(38,42)(43,48);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,43)(28,42)(29,44)(31,45)(32,46)(33,47)(39,48)(40,41);(1,3,6,13)(2,8,10,16)(4,14,5,12)(7,11,9,15)(19,31)(21,22)(24,39)(26,27)(29,40)(30,45)(32,33)(34,43)(35,37)(38,48)(41,44)(46,47)

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,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=ghi,f^c=fi,g^c=ghi,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, 239208);