#10349015

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

4584 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,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, 10256211);