#97296

order	512
nilpotency class	3
isoclinism	isoclinic class of #97250
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

464 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=ghi,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=eg,f^c=fgh

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, 4492);