#232164

order	512
nilpotency class	3
isoclinism	isoclinic class of #232153
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3184	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=efhi,d^2=eh,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139360);