#250491

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	#3546	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

935 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)(33,34,38,42)(35,39,45,53)(36,40,46,54)(37,48,49,41)(43,56,50,51)(44,58,59,52)(47,61,60,55)(57,63,62,64);(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)(35,43)(37,49)(39,51)(41,48)(44,57)(45,50)(47,60)(52,63)(53,56)(55,61)(58,64)(59,62);(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)(33,35)(34,39)(36,44)(37,50)(38,45)(40,52)(41,56)(42,53)(43,49)(46,59)(47,62)(48,51)(54,58)(55,64)(57,60)(61,63);(2,9)(5,19)(7,12)(8,23)(10,11)(14,18)(16,32)(20,27)(21,26)(22,28)(24,25)(29,31)(33,36)(34,40)(35,44)(37,47)(38,46)(39,52)(41,55)(42,54)(43,57)(45,59)(48,61)(49,60)(50,62)(51,63)(53,58)(56,64)

Polycyclic presentation:

2^9|a^2=i,c^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, 157687);