#232883

order	512
nilpotency class	3
isoclinism	isoclinic class of #232876
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	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1107 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)(4,9)(5,10)(6,11)(7,12)(8,15)(14,29)(16,24)(17,23)(18,22)(19,26)(20,28)(21,27)(25,30)(31,32)(33,34,39,44)(35,45,48,55)(36,49,52,58)(37,42,38,43)(40,53,56,47)(41,51,59,50)(46,54,63,57)(60,61,64,62);(1,3,7,17)(2,8,12,24)(4,18,20,31)(5,16,6,15)(9,25,28,29)(10,13,11,23)(14,21,30,19)(22,27,32,26)(33,35,39,48)(34,40,44,56)(36,46,52,63)(37,53,38,47)(41,57,59,54)(42,45,43,55)(49,60,58,64)(50,62,51,61);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,27)(12,28)(13,29)(15,18)(16,31)(17,30)(23,25)(24,32)(33,36)(34,41)(35,46)(37,50)(38,51)(39,52)(40,54)(42,58)(43,49)(44,59)(45,60)(47,61)(48,63)(53,62)(55,64)(56,57);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,26,28,27)(14,18,30,31)(22,25,32,29)(33,37,39,38)(34,42,44,43)(35,47,48,53)(36,50,52,51)(40,55,56,45)(41,58,59,49)(46,61,63,62)(54,64,57,60)

Polycyclic presentation:

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

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, 140079);