#588621

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=fg,e^2=gh,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, 495817);