#243105

order	512
nilpotency class	3
isoclinism	isoclinic class of #243097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

525 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)(22,30,32,31)(33,34,37,42,39,43,38,54)(35,44,46,56,48,60,47,40)(36,49,53,58,51,59,52,41)(45,55,50,64,61,63,62,57);(1,3,7,16)(2,8,12,23)(4,17,19,14)(5,15,6,21)(9,22,25,32)(10,13,11,27)(18,31,20,30)(24,29,26,28)(33,35,39,48)(34,40,43,56)(36,50,51,62)(37,47,38,46)(41,57,58,64)(42,60,54,44)(45,52,61,53)(49,63,59,55);(1,4,6,18,7,19,5,20)(2,9,11,24,12,25,10,26)(3,14,15,31,16,17,21,30)(8,22,13,28,23,32,27,29)(33,36,37,52,39,51,38,53)(34,41,42,59,43,58,54,49)(35,45,46,62,48,61,47,50)(40,55,44,57,56,63,60,64);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,19,20)(8,13,23,27)(9,24,25,26)(14,30,17,31)(22,29,32,28)(36,51)(41,58)(45,61)(49,59)(50,62)(52,53)(55,63)(57,64)

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,d^c=dh,f^c=fi,g^c=ghi

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, 150301);