#125277

order	512
nilpotency class	4
isoclinism	isoclinic class of #125098
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,7,8,5)(4,6)(10,14,15,16)(17,18,22,29,24,31,41,27,25,32,21,28,23,30,20,40)(19,33,36,45,38,48,42,26,39,44,35,46,37,43,34,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19,25,39)(18,26,32,47)(20,36,41,35)(21,42,22,34)(23,38,24,37)(27,33,40,44)(28,43,29,48)(30,46,31,45);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(17,20,23,21,25,41,24,22)(18,27,30,29,32,40,31,28)(19,34,37,35,39,42,38,36)(26,43,45,44,47,48,46,33)

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=cfi,d^a=df,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32473);