#300514

order	512
nilpotency class	3
isoclinism	isoclinic class of #300501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62492	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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:

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

Polycyclic presentation:

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

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, 207710);