#237919

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

786 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

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

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, 145115);