#188113

order	512
nilpotency class	3
isoclinism	isoclinic class of #188049
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3362 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 3362 conjugacy classes of subgroups — too many to draw. Only its 334 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)(3,7,8,5)(4,6)(9,10,14,20,15,19,13,18)(11,21,23,34,24,16,28,30)(12,17,25,31,26,32,27,33)(22,35,37,39,38,29,36,40);(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,29)(18,30)(19,21)(20,34)(25,36)(26,38)(27,37)(31,39)(32,35)(33,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,26)(10,17,19,32)(11,22,24,38)(13,27,14,25)(16,29,21,35)(18,33,20,31)(23,37,28,36)(30,40,34,39);(9,13,15,14)(10,18,19,20)(11,23,24,28)(12,25,26,27)(16,30,21,34)(17,31,32,33)(22,36,38,37)(29,39,35,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,g^c=gi

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, 95309);