#10218625

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	#39946	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=g,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, 10125821);