#7658816

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
prime radical	#36652	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

(2,4)(5,6,10,14)(7,15,16,11)(8,12,9,13)(17,20,18,19)(23,31)(28,38)(32,45)(33,46)(41,49)(42,50)(47,48)(51,52);(1,2,3,4)(7,16)(11,15)(17,18)(19,20)(22,27)(23,31)(29,39)(30,40)(32,45)(33,46)(43,44)(47,48);(5,7)(6,11)(8,17)(9,18)(10,16)(12,19)(13,20)(14,15)(21,22,26,27)(23,28,31,38)(24,29,35,39)(25,30,37,40)(32,41,45,49)(33,42,46,50)(34,43,36,44)(47,51,48,52);(5,6)(7,11)(8,12)(9,13)(10,14)(15,16)(17,19)(18,20)(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);(5,8,10,9)(6,12,14,13)(7,17,16,18)(11,19,15,20)(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);(5,9,10,8)(6,13,14,12)(7,18,16,17)(11,20,15,19)(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=g,c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566012);