#8057882

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965078);