#10218635

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,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, 10125831);