#300484

order	512
nilpotency class	3
isoclinism	isoclinic class of #300469
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11)(4,5)(6,9)(7,10)(8,13)(12,15)(14,16)(17,18,23,20)(19,27,30,42)(21,34,37,46)(22,33,35,26)(24,38,31,29)(25,40,32,36)(28,43,45,39)(41,47,48,44);(1,3,5,13,7,15,6,14)(2,8,9,11,10,16,4,12)(17,19,22,29,23,30,35,38)(18,24,26,27,20,31,33,42)(21,28,36,44,37,45,40,47)(25,39,34,41,32,43,46,48);(1,2,6,4,7,10,5,9)(3,12,13,8,15,11,14,16)(17,20,23,18)(19,24)(21,32,37,25)(22,33,35,26)(27,29)(28,39)(30,31)(34,36,46,40)(38,42)(41,44)(43,45)(47,48);(1,4,6,10,7,9,5,2)(3,12,14,16,15,11,13,8)(17,21,35,40,23,37,22,36)(18,25,33,46,20,32,26,34)(19,28,38,47,30,45,29,44)(24,39,42,48,31,43,27,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=eh,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=di,g^c=ghi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207680);