#98421

order	512
nilpotency class	3
isoclinism	isoclinic class of #98302
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

495 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5617);