#8757442

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	#45972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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)(4,6)(5,8)(10,14)(11,19)(12,22)(18,24)(25,26,31,28)(27,35,36,32)(29,39,42,33)(30,40,43,34)(37,44,49,47)(38,45,50,48)(41,46,53,52)(51,55,56,54);(1,2,4,6)(3,5,8,7)(9,10,15,14)(11,20,19,16)(12,17,22,23)(13,24,21,18)(27,36)(30,43)(32,35)(34,40)(37,49)(41,53)(44,47)(46,52);(1,2)(3,5)(4,6)(7,8)(9,11,15,19)(10,16,14,20)(12,21,22,13)(17,24,23,18)(25,27)(26,32)(28,35)(29,37)(30,38)(31,36)(33,44)(34,45)(39,47)(40,48)(41,51)(42,49)(43,50)(46,54)(52,55)(53,56);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,28,31,26)(27,32,36,35)(29,39,42,33)(30,40,43,34)(37,44,49,47)(38,45,50,48)(41,52,53,46)(51,54,56,55);(9,13,15,21)(10,18,14,24)(11,12,19,22)(16,17,20,23)(25,29)(26,33)(27,37)(28,39)(30,41)(31,42)(32,44)(34,46)(35,47)(36,49)(38,51)(40,52)(43,53)(45,54)(48,55)(50,56);(9,14,15,10)(11,20,19,16)(12,23,22,17)(13,24,21,18)(25,30,31,43)(26,34,28,40)(27,38,36,50)(29,41,42,53)(32,45,35,48)(33,46,39,52)(37,51,49,56)(44,54,47,55)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664638);