#330187

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

721 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 237383);