#125552

order	512
nilpotency class	4
isoclinism	isoclinic class of #125393
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=cfg,d^a=df,f^a=fi,c^b=chi,d^b=dg,e^b=efg,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, 32748);