#331296

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 663 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,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, 238492);