#10218573

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

4058 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gh,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, 10125769);