#8000973

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7908169);