#8057885

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965081);