#8417075

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	#52602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(4,6)(5,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26)(27,35)(28,33)(29,31)(30,34)(32,38)(36,49)(37,48)(39,47)(40,41)(42,44)(43,46)(45,51)(50,56)(52,55)(53,54);(1,2,4,6)(3,5,8,7)(10,14)(15,17)(16,21)(22,23)(26,29)(32,35)(33,41)(34,42)(45,49)(46,48)(47,54)(55,56);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,32)(28,36)(29,35)(30,43)(31,38)(33,45)(34,48)(37,44)(39,52)(40,51)(41,49)(42,46)(47,56)(50,53)(54,55);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,40)(26,33,29,41)(27,36,38,51)(30,39,44,53)(32,45,35,49)(34,47,42,54)(37,50,43,52)(46,55,48,56);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,26)(27,35,38,32)(28,33,40,41)(30,42,44,34)(36,45,51,49)(37,48,43,46)(39,47,53,54)(50,55,52,56);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,44)(26,34,29,42)(27,37,38,43)(28,39,40,53)(32,46,35,48)(33,47,41,54)(36,50,51,52)(45,55,49,56)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gi,e^2=gi,f^2=h,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, 8324271);