#125289

order	512
nilpotency class	4
isoclinism	isoclinic class of #125287
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32485);