#601600

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,d^2=h,e^2=fhi,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, 508796);