#588993

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=gh,e^2=fh,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, 496189);