#98133

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
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

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

normal subgroups only

This group has 460 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5329);