#7749870

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7657066);