#8416925

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	#54132	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8324121);