#125180

order	512
nilpotency class	4
isoclinism	isoclinic class of #125076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

432 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 110 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)(9,10,12,16,13,14,11,15)(17,18,21,26,23,28,22,27)(19,29,31,42,33,46,32,45)(20,34,36,25,38,43,37,44)(24,35,39,30,41,47,40,48);(1,3)(2,5)(4,8)(6,7)(10,14)(11,12)(17,19,23,33)(18,24,28,41)(20,35,38,47)(21,32,22,31)(25,42,44,45)(26,40,27,39)(29,34,46,43)(30,36,48,37);(9,11,13,12)(10,15,14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32376);