#10324601

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

4546 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10231797);