#10247621

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52644	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(1,2)(3,4)(5,7)(6,8)(25,26,31,29)(27,35,36,32)(28,39,42,33)(30,43,44,34)(37,45,40,47)(38,46,50,49)(41,48,54,52)(51,56,53,55);(1,3)(2,6)(4,7)(5,8)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,21,23)(26,29)(27,36)(33,39)(34,43)(37,40)(38,50)(48,52)(51,53);(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,24)(21,22)(25,27,31,36)(26,32,29,35)(28,40,42,37)(30,38,44,50)(33,47,39,45)(34,46,43,49)(41,53,54,51)(48,56,52,55);(1,2,5,7)(3,4,8,6)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,24,21)(25,28)(26,33)(27,37)(29,39)(30,41)(31,42)(32,45)(34,48)(35,47)(36,40)(38,51)(43,52)(44,54)(46,55)(49,56)(50,53);(1,4,5,6)(2,3,7,8)(9,11)(10,12)(13,21)(14,19)(15,17)(16,23)(18,22)(20,24)(25,29,31,26)(27,35,36,32)(28,39,42,33)(30,43,44,34)(37,47,40,45)(38,49,50,46)(41,52,54,48)(51,56,53,55);(1,3)(2,4)(5,8)(6,7)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,30,31,44)(26,34,29,43)(27,38,36,50)(28,41,42,54)(32,46,35,49)(33,48,39,52)(37,51,40,53)(45,55,47,56)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154817);