#601055

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=h,e^2=fi,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, 508251);