#10349121

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,e^2=h,f^2=h,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, 10256317);