#4503035

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1570 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)(7,13)(8,15)(11,16)(12,17)(21,22,25,26)(23,29,30,27)(24,32,33,28)(31,34,36,35)(37,38,41,45)(39,43,47,42)(40,48,50,44)(46,51,49,52);(6,10)(11,16)(12,17)(18,20)(22,26)(23,30)(24,33)(34,35)(37,39,41,47)(38,42,45,43)(40,46,50,49)(44,51,48,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35)(37,39)(38,43)(40,49)(41,47)(42,45)(44,51)(46,50)(48,52);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36)(37,40,41,50)(38,44,45,48)(39,46,47,49)(42,51,43,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)(37,38)(39,43)(40,44)(41,45)(42,47)(46,52)(48,50)(49,51)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=g,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4410231);