#9913327

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	C₂×#562	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(11,17)(14,21)(18,23)(22,24)(26,31)(29,41)(32,35)(33,42)(37,51)(40,52)(44,47)(50,56);(2,6)(3,7)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(28,39)(33,42)(36,49)(40,52)(44,47)(46,53)(50,56)(54,55);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(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);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(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);(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(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);(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(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=g,b^2=gi,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820523);