#605413

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1566 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=hi,d^2=i,e^2=fgi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 512609);