#138195

order	512
nilpotency class	4
isoclinism	isoclinic class of #138195
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=fi

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, 45391);