#124709

order	512
nilpotency class	4
isoclinism	isoclinic class of #124708
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,7,8,5)(4,6)(9,10,13,19,14,20,23,33,15,21,31,40,32,18,37,39)(11,22,12,26,24,16,28,34,25,35,27,38,30,36,29,17);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,21,36)(12,23,27,37)(13,29,31,28)(14,30,32,24)(17,33,34,39)(18,35,20,22)(19,38,40,26);(9,12)(10,17,21,34)(11,23)(13,30)(14,28)(15,27)(16,33,36,39)(18,38,20,26)(19,35,40,22)(24,31)(25,37)(29,32)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=g,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 31905);