#8416923

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4108 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gi,e^2=i,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, 8324119);