#8416922

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	#52418	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,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, 8324118);