#6132632

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

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=h,e^2=gh,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, 6039828);