#232665

order	512
nilpotency class	3
isoclinism	isoclinic class of #232654
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	#3192	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1123 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)(3,13)(4,18)(5,12)(6,11)(7,10)(8,16)(9,21)(14,23)(15,24)(17,25)(19,32)(20,27)(22,28)(26,30)(29,31)(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)(4,19)(5,17)(6,16)(7,15)(9,26)(10,25)(11,24)(12,13)(14,20)(18,31)(21,29)(22,30)(23,28)(27,32)(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)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,27)(11,28)(12,18)(13,26)(15,29)(16,19)(17,30)(24,32)(25,31)(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);(2,10)(3,15)(9,27)(11,12)(14,29)(16,17)(18,28)(19,30)(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|b^2=i,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139861);