#8417181

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=i,e^2=gi,f^2=g,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, 8324377);