#294998

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 150 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)(11,16)(12,23)(13,19)(14,18)(15,20)(17,21)(22,24)(25,26,30,34)(27,31,36,44)(28,38,41,53)(29,42,39,33)(32,46,47,40)(35,48,50,56)(37,52,49,45)(43,54,55,51);(2,5)(6,7)(9,11,15,21)(10,16,20,17)(12,13,22,14)(18,23,19,24)(25,27,30,36)(26,31,34,44)(28,35,41,50)(29,37,39,49)(32,43,47,55)(33,45,42,52)(38,48,53,56)(40,51,46,54);(9,12,14,11,15,22,13,21)(10,17,19,24,20,16,18,23)(25,28,39,46,30,41,29,40)(26,32,42,53,34,47,33,38)(27,35,49,54,36,50,37,51)(31,43,52,56,44,55,45,48);(1,3,4,8)(2,6,5,7)(11,21)(12,22)(16,17)(23,24)(27,36)(31,44)(35,50)(37,49)(43,55)(45,52)(48,56)(51,54)

Polycyclic presentation:

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

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, 202194);