#4500988

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

1554 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)(7,11)(8,15)(9,10)(12,17)(13,20)(14,18)(16,19)(21,22,26,27)(23,31,32,28)(24,34,35,29)(25,33,36,30)(37,38)(39,45)(40,48)(41,44)(42,47)(43,49)(46,50)(51,52);(6,10)(11,18)(12,15)(19,20)(22,27)(23,32)(24,35)(25,36)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,7,9,14)(6,11,10,18)(8,16,17,13)(12,20,15,19)(21,23,26,32)(22,28,27,31)(24,30,35,33)(25,34,36,29)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2,3,4)(5,8)(6,12)(7,13)(9,17)(10,15)(11,19)(14,16)(18,20)(21,24,26,35)(22,29,27,34)(23,33,32,30)(25,28,36,31)(37,39)(38,42)(40,46)(41,47)(43,50)(44,45)(48,51)(49,52);(5,6)(7,11)(8,12)(9,10)(13,19)(14,18)(15,17)(16,20)(21,25,26,36)(22,30,27,33)(23,34,32,29)(24,28,35,31)(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=fgi,e^2=g,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, 4408184);