#8324605

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231801);