#6132640

order	512
nilpotency class	2
isoclinism	isoclinic class of #6131635
automorphism	Aut(G) order 7340032
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1732 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1732 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|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, 6039836);