#113642

order	512
nilpotency class	3
isoclinism	isoclinic class of #113634
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1719 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1719 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,g^b=ghi,d^c=di,e^c=ei,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, 20838);