#8754862

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	#56209	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(28,39)(33,42)(36,49)(40,52)(44,47)(46,53)(50,56)(54,55);(3,7)(5,8)(9,10,14,13)(11,18,17,15)(12,20,21,16)(19,22,23,24)(26,31)(28,39)(29,41)(32,35)(36,49)(37,51)(46,53)(54,55);(1,2)(3,5)(4,6)(7,8)(9,11,14,17)(10,15,13,18)(12,19,21,23)(16,22,20,24)(25,26,30,31)(27,35,38,32)(28,33,39,42)(29,34,41,43)(36,47,49,44)(37,48,51,45)(40,46,52,53)(50,55,56,54);(9,10,14,13)(11,15,17,18)(12,16,21,20)(19,22,23,24)(25,27,30,38)(26,32,31,35)(28,36,39,49)(29,37,41,51)(33,44,42,47)(34,45,43,48)(40,50,52,56)(46,54,53,55);(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)(26,33)(27,36)(29,40)(30,39)(31,42)(32,44)(34,46)(35,47)(37,50)(38,49)(41,52)(43,53)(45,54)(48,55)(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)(28,40)(30,41)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(39,52)(42,53)(44,54)(47,55)(49,56)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=i,e^2=g,f^2=g,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, 8662058);