#250499

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	#61795	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

955 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)(33,34,39,43)(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)(2,7)(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)(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);(1,4,6,17)(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)(35,45)(37,38)(40,54)(42,52)(44,56)(46,60)(47,48)(49,58)(50,53)(55,63)(59,64)(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)(33,36)(34,41)(35,46)(37,53)(38,50)(39,51)(40,55)(42,58)(43,57)(44,59)(45,60)(47,62)(48,61)(49,52)(54,63)(56,64)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,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, 157695);