#8926712

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	#57725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=g,e^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, 8833908);