#129335

order	512
nilpotency class	4
isoclinism	isoclinic class of #12715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

912 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fi,f^d=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, 36531);