#5475831

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

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, 5383027);