#96994

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

490 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg

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, 4190);