#8093615

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	C₂⁴	C₂⁵
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)(3,7)(4,5)(6,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,31,29)(27,32,38,36)(28,39,42,33)(30,43,44,34)(35,47,40,45)(37,50,51,46)(41,48,54,52)(49,55,53,56);(2,4)(6,7)(9,11,14,10)(12,15,20,17)(13,16,21,18)(19,23,24,22)(26,29)(32,36)(33,39)(34,43)(45,47)(46,50)(48,52)(55,56);(1,3)(2,6)(4,7)(5,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27,31,38)(26,32,29,36)(28,40,42,35)(30,37,44,51)(33,47,39,45)(34,46,43,50)(41,53,54,49)(48,56,52,55);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,28,31,42)(26,33,29,39)(27,35,38,40)(30,41,44,54)(32,45,36,47)(34,48,43,52)(37,49,51,53)(46,55,50,56);(25,29,31,26)(27,36,38,32)(28,39,42,33)(30,43,44,34)(35,47,40,45)(37,50,51,46)(41,52,54,48)(49,56,53,55);(25,30)(26,34)(27,37)(28,41)(29,43)(31,44)(32,46)(33,48)(35,49)(36,50)(38,51)(39,52)(40,53)(42,54)(45,55)(47,56)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000811);