#285384

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192580);