#125203

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 276 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(2,5,6,7)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,29,35,30,36,45,33,31,37,46,53,44,52,28,34)(27,38,40,51,41,54,39,55,42,48,43,32,47,49,56,50);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,42)(26,32,37,51)(28,43,45,40)(29,39,46,56)(30,47,44,41)(33,49,34,54)(35,38,53,48)(36,55,52,50);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,44,46,31,45,30,29)(26,33,52,35,37,34,36,53)(27,39,47,40,42,56,41,43)(32,48,55,54,51,38,50,49)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fhi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ci,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, 32399);