#300512

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	#18430	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 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:

(1,2,7,4)(3,11,15,26)(5,18,20,30)(6,17,21,10)(8,22,12,14)(9,24,16,19)(13,27,29,23)(25,31,32,28)(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);(1,3,6,14,7,15,21,22)(2,8,10,11,4,12,17,26)(5,13,19,28,20,29,24,31)(9,23,18,25,16,27,30,32)(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);(1,4,7,2)(3,12)(5,9,20,16)(6,17,21,10)(8,15)(11,22)(13,23)(14,26)(18,24,30,19)(25,28)(27,29)(31,32)(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);(1,5,6,19,7,20,21,24)(2,9,10,18,4,16,17,30)(3,13,14,28,15,29,22,31)(8,23,11,25,12,27,26,32)(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=fh,d^2=e,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, 207708);