#333156

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,13,8,14,6,12)(3,15,19,9,20,28,18,29)(4,10,25,31,5,23,22,11)(16,30,24,26,17,27,21,32)(33,34,38,43,40,44,39,56)(35,45,47,58,36,48,50,41)(37,51,53,61,55,42,54,60)(46,62,52,57,49,59,63,64);(1,3,8,20)(2,9,14,29)(4,21,5,24)(6,19,7,18)(10,30,23,27)(11,26,31,32)(12,28,13,15)(16,25,17,22)(33,35,40,36)(34,41,44,58)(37,52,55,63)(38,50,39,47)(42,59,51,62)(43,48,56,45)(46,54,49,53)(57,61,64,60);(1,4,8,5)(2,10,14,23)(3,16,20,17)(6,22,7,25)(9,26,29,32)(11,13,31,12)(15,30,28,27)(18,21,19,24)(33,36,40,35)(34,41,44,58)(37,49,55,46)(38,50,39,47)(42,57,51,64)(43,45,56,48)(52,53,63,54)(59,61,62,60);(1,5,8,4)(2,11,14,31)(3,17,20,16)(6,22,7,25)(9,27,29,30)(10,13,23,12)(15,32,28,26)(18,21,19,24)(33,37,40,55)(34,42,44,51)(35,46,36,49)(38,54,39,53)(41,57,58,64)(43,61,56,60)(45,62,48,59)(47,63,50,52)

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

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, 240352);