#125646

order	512
nilpotency class	4
isoclinism	isoclinic class of #125644
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×#97	C₂³
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

506 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=df,e^b=egi,f^b=fi,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213794
maximal normal subgroup of	#72213794
maximal quotient of	#72213167, #72213174, #72213261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32842);