#114749

order	512
nilpotency class	3
isoclinism	isoclinic class of #114746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21945);