#168096

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

674 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

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, 75292);