#330832

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

480 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)(17,18,22,26,21,25,30,32)(19,27,28,24,20,23,29,31)(33,34)(35,38)(36,39)(37,40)(41,48)(42,46)(43,45)(44,47);(2,6)(7,13)(8,14)(15,16)(17,19)(18,23)(20,21)(22,28)(24,32)(25,27)(26,31)(29,30)(33,35,36,42)(34,38,39,46)(37,41,44,43)(40,48,47,45);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20,21,19)(18,24,25,31)(22,28,30,29)(23,26,27,32)(34,39)(35,41)(37,44)(38,45)(42,43)(46,48);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(18,25)(19,20)(26,32)(28,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=e,d^2=eh,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, 238028);