#8115684

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

3929 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gi,e^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, 8022880);