#140692

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

375 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,d^c=di,f^c=fi,f^d=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, 47888);