#8926774

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	#56809	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833970);