#331491

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

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

656 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,6)(4,5)(7,8)(9,10,12,17,13,16,20,15)(11,14,19,22,18,23,24,21)(25,26)(27,31)(28,32)(29,34)(30,33)(35,43)(36,45)(37,44)(38,46)(39,48)(40,47)(41,50)(42,49)(51,54)(52,56)(53,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27,29,36,30,37,42,53)(26,31,34,45,33,44,49,55)(28,38,39,51,40,52,41,35)(32,46,48,54,47,56,50,43);(3,7)(6,8)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(25,28,30,40)(26,32,33,47)(27,35,37,51)(29,41,42,39)(31,43,44,54)(34,50,49,48)(36,52,53,38)(45,56,55,46);(1,3,4,7)(2,6,5,8)(10,16)(11,18)(15,17)(19,24)(26,33)(31,44)(32,47)(34,49)(43,54)(45,55)(46,56)(48,50)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,d^c=de,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, 238687);