#215216

order	512
nilpotency class	3
isoclinism	isoclinic class of #162550
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1014 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122412);