#316986

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 150 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)(4,5)(6,8)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24)(25,26)(27,35)(28,33)(29,32)(30,34)(31,37)(36,39)(38,40);(2,5)(6,7)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,23)(20,24)(25,27,30,38,28,37,29,36)(26,31,34,39,33,35,32,40);(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,23)(21,24)(27,37)(31,35)(36,38)(39,40)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,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, 224182);