#330840

order	512
nilpotency class	3
isoclinism	isoclinic class of #329551
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=e,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 238036);