#113407

order	512
nilpotency class	3
isoclinism	isoclinic class of #113404
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

760 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)(3,11,13,7)(4,8)(5,18)(6,9)(10,21)(12,22,30,29)(14,23,19,27)(15,32,31,24)(16,28)(17,25)(20,26)(33,34)(35,43,46,56)(36,47,49,40)(37,41)(38,42)(39,45,54,50)(44,59)(48,57,51,55)(52,58)(53,60)(61,64)(62,63);(1,3,6,13)(2,7,9,11)(4,15,17,31)(5,19,21,14)(8,24,25,32)(10,27,18,23)(12,20,30,16)(22,28,29,26)(33,35)(34,39)(36,44)(37,50)(38,46)(40,53)(41,56)(42,54)(43,58)(45,52)(47,63)(48,62)(49,61)(51,60)(55,59)(57,64);(1,4,6,17)(2,8,9,25)(3,12,13,30)(5,20,21,16)(7,22,11,29)(10,28,18,26)(14,31,19,15)(23,32,27,24)(33,36)(34,40)(35,44)(37,51)(38,49)(39,53)(41,57)(42,47)(43,59)(45,62)(46,61)(48,52)(50,60)(54,63)(55,58)(56,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,f^c=fh,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, 20603);