#125900

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1954 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,h^a=hi,d^b=df,e^b=eg,d^c=dg,e^c=eh,h^c=hi

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, 33096);