#243096

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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)(3,13,16,27)(4,9)(5,10)(6,11)(7,12)(8,15,23,21)(14,29,17,28)(18,24)(19,25)(20,26)(22,31,32,30)(33,34,37,42,39,43,38,54)(35,44,46,56,48,60,47,40)(36,49,52,59,53,58,51,41)(45,55,50,64,62,63,61,57);(1,3,7,16)(2,8,12,23)(4,17,20,14)(5,15,6,21)(9,22,26,32)(10,13,11,27)(18,31,19,30)(24,28,25,29)(33,35,39,48)(34,40,43,56)(36,50,53,61)(37,47,38,46)(41,57,59,64)(42,60,54,44)(45,51,62,52)(49,63,58,55);(1,4,6,19,7,20,5,18)(2,9,11,25,12,26,10,24)(3,14,15,30,16,17,21,31)(8,22,13,28,23,32,27,29)(33,36,37,51,39,53,38,52)(34,41,42,58,43,59,54,49)(35,45,46,61,48,62,47,50)(40,55,44,57,56,63,60,64);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,13,23,27)(9,24,26,25)(14,30,17,31)(22,28,32,29)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150292);