#113909

order	512
nilpotency class	3
isoclinism	isoclinic class of #113905
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

754 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:

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

Polycyclic presentation:

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

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, 21105);