#250509

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 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)(4,8)(5,9)(6,11)(7,14)(10,19)(13,28)(15,26)(16,23)(17,25)(18,21)(20,31)(22,29)(24,32)(27,30)(33,34,39,35)(36,40,48,43)(37,41,50,44)(38,45,53,42)(46,52,55,49)(47,57,60,54)(51,58,62,56)(59,61,64,63);(1,3)(2,7)(4,13)(5,18)(6,15)(8,22)(9,12)(10,26)(11,21)(14,19)(16,31)(17,30)(20,25)(23,28)(24,27)(29,32)(33,35,39,34)(36,46,48,55)(37,44,50,41)(38,42,53,45)(40,49,43,52)(47,59,60,64)(51,56,62,58)(54,61,57,63);(1,4,6,17)(2,8,11,25)(3,13,15,30)(5,16,19,32)(7,20,21,22)(9,23,10,24)(12,27,26,28)(14,29,18,31)(33,36)(34,40)(35,43)(37,47)(38,52)(39,48)(41,54)(42,46)(44,57)(45,55)(49,53)(50,60)(51,63)(56,59)(58,64)(61,62);(2,9)(5,19)(7,12)(8,23)(10,11)(14,18)(16,32)(20,27)(21,26)(22,28)(24,25)(29,31)(33,37,39,50)(34,41,35,44)(36,47,48,60)(38,51,53,62)(40,54,43,57)(42,56,45,58)(46,59,55,64)(49,61,52,63)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157705);