#8757410

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,e^2=ghi,f^2=h,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, 8664606);