#9040125

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	#41385	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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)(11,15)(12,18)(13,21)(14,17)(16,22)(19,24)(20,23)(26,31)(27,35)(29,42)(33,38)(36,39)(40,54)(46,51)(55,56);(2,6)(3,7)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(26,31)(32,44)(33,38)(34,43)(45,47)(46,51)(48,52)(55,56);(1,3)(2,4)(5,7)(6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,26,30,31)(27,32,35,44)(28,38,41,33)(29,43,42,34)(36,47,39,45)(37,51,49,46)(40,48,54,52)(50,55,53,56);(1,4)(2,3)(5,8)(6,7)(25,27,30,35)(26,32,31,44)(28,39,41,36)(29,37,42,49)(33,47,38,45)(34,46,43,51)(40,53,54,50)(48,56,52,55);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,28,30,41)(26,33,31,38)(27,36,35,39)(29,40,42,54)(32,45,44,47)(34,48,43,52)(37,50,49,53)(46,55,51,56);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29)(26,34)(27,37)(28,40)(30,42)(31,43)(32,46)(33,48)(35,49)(36,50)(38,52)(39,53)(41,54)(44,51)(45,55)(47,56)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,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, 8947321);