#352293

order	512
nilpotency class	3
isoclinism	isoclinic class of #352272
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

591 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)(3,13)(4,18)(5,12)(6,9)(7,11)(8,15)(10,19)(14,28)(16,24)(17,22)(20,26)(21,25)(23,30)(27,29)(31,32)(33,34,38,42)(35,39,44,52)(36,46,49,61)(37,50,47,41)(40,54,55,48)(43,56,58,64)(45,60,57,53)(51,62,63,59);(1,3,7,17,6,16,5,15)(2,8,12,24,9,22,11,13)(4,14,21,31,20,30,19,29)(10,23,26,32,25,28,18,27)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,53)(42,52)(46,56)(47,57)(48,59)(49,58)(50,60)(54,62)(55,63)(61,64);(2,9)(8,22)(10,25)(11,12)(13,24)(18,26)(23,28)(27,32)(33,36,47,54,38,49,37,48)(34,40,50,61,42,55,41,46)(35,43,57,62,44,58,45,59)(39,51,60,64,52,63,53,56);(1,4)(2,10)(3,14)(5,19)(6,20)(7,21)(8,23)(9,25)(11,18)(12,26)(13,27)(15,29)(16,30)(17,31)(22,28)(24,32)(35,44)(39,52)(43,58)(45,57)(51,63)(53,60)(56,64)(59,62)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fhi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=di,g^d=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259489);