#5511059

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

1532 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(37,38)(39,43)(40,49)(41,48)(42,47)(44,50)(45,51)(46,52);(6,7)(9,17)(11,13)(15,20)(23,29)(26,33)(30,35)(34,36)(37,39,42,50)(38,43,47,44)(40,48,51,46)(41,45,52,49);(5,7,10,6)(8,13,16,11)(9,12,17,14)(15,18,20,19)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35)(37,39)(38,44)(40,46)(41,49)(42,50)(43,47)(45,52)(48,51);(1,2,3,4)(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,45)(39,48)(41,44)(42,51)(43,52)(46,50)(47,49);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41)(38,46)(39,49)(40,44)(42,52)(43,51)(45,50)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=f,d^2=hi,e^2=gh,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, 5418255);