#97365

order	512
nilpotency class	3
isoclinism	isoclinic class of #97316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2819	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

520 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=hi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ehi,f^c=fhi

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, 4561);