#115530

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

749 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)(3,12)(4,17,20,9)(5,21)(6,10)(7,11)(8,14)(13,25,30,29)(15,24)(16,28)(18,32,31,23)(19,26,22,27)(33,34,38,42,39,43,37,50)(35,40,45,55,46,56,51,54)(36,41,49,60,48,58,52,59)(44,53,62,57,61,64,47,63);(1,3,7,16)(2,8,11,24)(4,18,20,31)(5,15,6,14)(9,25,17,29)(10,28,21,12)(13,22,30,19)(23,26,32,27)(33,35)(34,40)(36,47)(37,51)(38,45)(39,46)(41,57)(42,55)(43,56)(44,49)(48,62)(50,54)(52,61)(53,59)(58,63)(60,64);(1,4,7,20)(2,9,11,17)(3,13,16,30)(5,19,6,22)(8,23,24,32)(10,26,21,27)(12,29,28,25)(14,18,15,31)(33,36)(34,41)(35,44)(37,52)(38,49)(39,48)(40,53)(42,60)(43,58)(45,62)(46,61)(47,51)(50,59)(54,63)(55,57)(56,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,g^c=gh

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, 22726);