#10312625

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

4353 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10219821);