#113588

order	512
nilpotency class	3
isoclinism	isoclinic class of #113584
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60967	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

762 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)(3,13,16,8)(4,17,21,27)(5,11)(6,12)(7,10)(9,19,26,20)(14,30)(15,28,22,24)(18,23)(25,31)(29,32)(33,34,38,42,39,44,37,43)(35,45,47,40,48,55,53,56)(36,41,51,58,52,59,50,60)(46,57,61,54,62,64,49,63);(1,3)(2,8)(4,18)(5,15)(6,22)(7,16)(9,25)(10,13)(11,24)(12,28)(14,20)(17,29)(19,31)(21,32)(23,27)(26,30)(33,35)(34,40)(36,49)(37,53)(38,47)(39,48)(41,57)(42,55)(43,45)(44,56)(46,51)(50,62)(52,61)(54,58)(59,64)(60,63);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,23)(10,26)(11,27)(12,17)(13,29)(15,32)(16,31)(18,22)(24,30)(25,28)(33,36)(34,41)(35,46)(37,50)(38,51)(39,52)(40,54)(42,58)(43,60)(44,59)(45,57)(47,61)(48,62)(49,53)(55,64)(56,63)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20784);