#6132642

order	512
nilpotency class	2
isoclinism	isoclinic class of #6131635
automorphism	Aut(G) order 25165824
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1262 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 490 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 — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6039838);