#250500

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

963 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)(4,9)(5,17)(6,19)(7,11)(8,13)(10,16)(14,22)(15,24)(18,25)(20,23)(21,26)(27,29)(28,31)(30,32)(33,34,39,43)(35,44,47,54)(36,49,51,58)(37,52,38,42)(40,48,56,45)(41,53,57,50)(46,55,62,64)(59,60,63,61);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,24)(10,22)(11,23)(12,19)(16,28)(17,27)(18,30)(21,29)(25,31)(26,32)(33,35)(34,40)(36,46)(37,48)(38,45)(39,47)(41,55)(42,54)(43,56)(44,52)(49,59)(50,60)(51,62)(53,61)(57,64)(58,63);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,18,21)(8,12,23,24)(10,25,26,17)(14,28,30,29)(22,27,32,31)(35,45)(37,38)(40,54)(42,52)(44,56)(46,60)(47,48)(49,58)(50,53)(55,63)(59,64)(61,62);(1,5,7,18)(2,10,11,26)(3,14,15,30)(4,16,6,21)(8,22,23,32)(9,25,19,17)(12,27,24,31)(13,28,20,29)(33,36,39,51)(34,41,43,57)(35,46,47,62)(37,53,38,50)(40,55,56,64)(42,58,52,49)(44,59,54,63)(45,60,48,61)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=h,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, 157696);