#303935

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

799 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,23,31,40)(20,32,35,44)(21,26,34,25)(24,37,42,36)(28,45,46,41)(29,43,30,39)(33,38,47,48);(1,3,5,4)(2,6,8,7)(11,16)(12,15)(17,19)(18,23)(20,33)(21,29)(22,31)(24,41)(25,39)(26,43)(27,40)(28,37)(30,34)(32,38)(35,47)(36,46)(42,45)(44,48);(1,3,5,4)(2,7,8,6)(9,11,15,10,13,16,12,14)(17,20,34,37,22,35,21,36)(18,24,26,44,27,42,25,32)(19,28,30,47,31,46,29,33)(23,38,43,41,40,48,39,45);(1,4,5,3)(2,7,8,6)(20,35)(24,42)(28,46)(32,44)(33,47)(36,37)(38,48)(41,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=efh,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211131);