#305022

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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

629 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=ef,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212218);