#499429

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

618 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406625);