#232819

order	512
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60953	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1109 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)(5,10)(6,11)(7,12)(8,16)(9,19)(14,25)(15,24)(17,23)(20,26)(21,27)(22,31)(28,30)(29,32)(33,34,39,44)(35,40,47,56)(36,48,50,59)(37,43,38,42)(41,52,58,51)(45,57,61,64)(46,60,53,55)(49,54,62,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,25)(10,13)(11,23)(12,24)(18,28)(19,29)(20,31)(21,30)(22,27)(26,32)(33,35)(34,40)(36,49)(37,46)(38,53)(39,47)(41,57)(42,55)(43,60)(44,56)(45,52)(48,63)(50,62)(51,61)(54,59)(58,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,18)(12,27)(13,28)(15,29)(16,30)(17,31)(23,32)(24,25)(33,36)(34,41)(35,45)(37,52)(38,51)(39,50)(40,54)(42,59)(43,48)(44,58)(46,62)(47,61)(49,53)(55,64)(56,63)(57,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,13,24,23)(9,26,27,18)(14,29,31,30)(22,28,25,32)

Polycyclic presentation:

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

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, 140015);