#5021432

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3290 conjugacy classes of subgroups; only the 634 normal subgroups are shown.

normal subgroups only

This group has 3290 conjugacy classes of subgroups — too many to draw. Only its 634 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

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, 4928628);