#10217782

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	#49747	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,34)(28,33)(29,37)(30,35)(31,36)(32,38)(39,40);(1,2)(3,7)(4,6)(5,8)(12,19)(16,22)(18,23)(20,24)(25,27)(26,32)(28,35)(29,39)(30,37)(31,38)(33,40)(34,36);(1,3)(2,5)(4,8)(6,7)(10,15)(11,17)(16,22)(18,23)(25,28,31,29)(26,33,36,37)(27,35,38,39)(30,32,40,34);(2,6)(5,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,29,31,28)(26,33,36,37)(27,35,38,39)(30,34,40,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,27)(26,34)(28,35)(29,39)(30,33)(31,38)(32,36)(37,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30)(26,35)(27,37)(28,32)(29,34)(31,40)(33,38)(36,39)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10124978);