#5477955

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fgh,d^2=h,e^2=g,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, 5385151);