#232163

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	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

407 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)(4,15,18,25)(5,11)(6,10)(8,19,26,30)(9,17)(12,24)(14,22,20,27)(16,21)(28,32)(29,31)(33,34)(35,44)(36,41,50,58)(37,51,49,63)(38,53)(39,43)(40,47)(42,52)(45,57,61,64)(46,62,60,55)(48,59)(54,56);(1,3,5,13)(2,7,11,23)(4,16,18,29)(6,20,17,14)(8,24,26,32)(9,22,10,27)(12,19,28,30)(15,21,25,31)(33,35,39,48)(34,40,43,56)(36,46,50,60)(37,45,49,61)(38,54,52,47)(41,57,58,64)(42,59,53,44)(51,55,63,62);(1,4,17,30,5,18,6,19)(2,8,10,15,11,26,9,25)(3,12,20,16,13,28,14,29)(7,21,22,24,23,31,27,32)(33,36,38,49,39,50,52,37)(34,41,53,63,43,58,42,51)(35,45,47,60,48,61,54,46)(40,55,59,64,56,62,44,57);(2,9,11,10)(3,13)(7,22,23,27)(8,25,26,15)(12,28)(14,20)(16,29)(21,24,31,32)(33,37,52,50,39,49,38,36)(34,41,53,63,43,58,42,51)(35,46,54,61,48,60,47,45)(40,55,59,64,56,62,44,57)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efi,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, 139359);