#10368027

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

4506 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=i,e^2=gh,f^2=gh,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, 10275223);