#8417333

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	#36032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=hi,e^2=hi,f^2=gh,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, 8324529);