#232495

order	512
nilpotency class	3
isoclinism	isoclinic class of #232487
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	#3186	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1117 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)(3,12)(4,17)(5,22)(6,10)(7,11)(8,15)(9,20)(13,30)(14,25)(16,24)(18,29)(19,28)(21,27)(23,31)(26,32)(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)(5,16)(6,15)(7,14)(9,26)(10,25)(11,24)(12,22)(13,19)(17,23)(20,31)(21,32)(27,30)(28,29)(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)(3,13)(5,19)(6,20)(7,21)(8,23)(10,27)(11,28)(12,29)(14,31)(15,18)(16,32)(17,22)(24,30)(25,26)(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);(3,14)(8,24)(12,25)(13,31)(15,16)(18,32)(23,30)(26,29)(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,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fh,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, 139691);