#333185

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

575 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=hi,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, 240381);