#8115412

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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)(3,5)(4,7)(6,8)(9,10,15,12)(11,18,14,16)(13,20,22,17)(19,23,21,24)(25,26,31,32)(27,28,38,33)(29,41,42,34)(30,44,45,35)(36,46,49,39)(37,47,51,40)(43,48,54,53)(50,52,56,55);(2,3)(6,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(26,32)(28,33)(29,42)(35,44)(36,49)(40,47)(43,54)(50,56);(1,2,5,3)(4,6,8,7)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,27,31,38)(26,33,32,28)(29,36,42,49)(30,37,45,51)(34,46,41,39)(35,47,44,40)(43,50,54,56)(48,55,53,52);(1,3,5,2)(4,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,28,31,33)(26,27,32,38)(29,39,42,46)(30,40,45,47)(34,36,41,49)(35,37,44,51)(43,52,54,55)(48,50,53,56);(1,4,5,8)(2,6,3,7)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,29,31,42)(26,34,32,41)(27,36,38,49)(28,39,33,46)(30,43,45,54)(35,48,44,53)(37,50,51,56)(40,52,47,55);(25,30,31,45)(26,35,32,44)(27,37,38,51)(28,40,33,47)(29,43,42,54)(34,48,41,53)(36,50,49,56)(39,52,46,55)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022608);