#242376

order	512
nilpotency class	3
isoclinism	isoclinic class of #242365
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
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

577 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gi,e^c=eh,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, 149572);