#391895

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 290 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299091);