#332010

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	#3214	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

512 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)(19,29,32,37)(20,26,35,41)(21,24,22,27)(25,36,40,31)(30,44,46,43)(33,42,34,38)(39,47,48,45);(1,2)(3,8)(4,9)(5,7)(6,10)(11,12)(13,16)(14,15)(18,24)(20,33)(21,22)(25,29)(26,41)(27,28)(30,45)(31,36)(34,35)(37,40)(43,44)(46,47);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,19)(18,25)(20,30)(21,31)(22,36)(23,32)(24,37)(26,39)(27,29)(28,40)(33,45)(34,47)(35,46)(38,44)(41,48)(42,43);(1,3)(2,8)(4,14)(5,12)(6,13)(7,11)(9,15)(10,16)(17,20,23,35)(18,26,28,41)(19,30,32,46)(21,34,22,33)(24,38,27,42)(25,39,40,48)(29,43,37,44)(31,47,36,45)

Polycyclic presentation:

2^9|a^2=h,d^2=h,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, 239206);