#232559

order	512
nilpotency class	3
isoclinism	isoclinic class of #232546
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1124 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139755);