#316936

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gi

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, 224132);