#8115305

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	#34636	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:

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

Polycyclic presentation:

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

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, 8022501);