#382708

order	512
nilpotency class	3
isoclinism	isoclinic class of #190028
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1898 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1898 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289904);