#244394

order	512
nilpotency class	3
isoclinism	isoclinic class of #233501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151590);