#329661

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

635 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236857);