#8115405

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8022601);