#125081

order	512
nilpotency class	4
isoclinism	isoclinic class of #125076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	C₂×#466	C₄
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

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

normal subgroups only

This group has 399 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,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, 32277);