#7659023

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	#37454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6362 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)(6,10)(7,13)(12,17)(14,19)(21,22,27,25)(23,31,34,28)(24,29,35,36)(26,30,39,38)(32,43,45,40)(33,44,47,41)(37,42,48,49)(46,51,52,50);(1,2,3,4)(6,10)(11,16)(12,17)(18,20)(24,35)(29,36)(32,45)(37,48)(40,43)(42,49)(46,52)(50,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,27,34)(22,28,25,31)(24,32,35,45)(26,33,39,47)(29,40,36,43)(30,41,38,44)(37,46,48,52)(42,50,49,51);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,29)(23,32)(25,36)(26,37)(27,35)(28,40)(30,42)(31,43)(33,46)(34,45)(38,49)(39,48)(41,50)(44,51)(47,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,27,22)(23,31,34,28)(24,36,35,29)(26,30,39,38)(32,43,45,40)(33,41,47,44)(37,42,48,49)(46,50,52,51);(21,26)(22,30)(23,33)(24,37)(25,38)(27,39)(28,41)(29,42)(31,44)(32,46)(34,47)(35,48)(36,49)(40,50)(43,51)(45,52)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,e^2=h,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, 7566219);