#8000969

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

4699 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)(9,10,15,12)(11,17,18,13)(14,20,22,16)(19,21,24,23)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,17)(14,19)(15,18)(16,21)(20,23)(22,24)(25,27,31,30)(26,32,28,34)(29,37,38,35)(33,40,36,39);(1,3,4,7)(2,6,5,8)(10,12)(11,18)(14,22)(21,23)(26,28)(32,34)(33,36)(39,40);(9,12,15,10)(11,17,18,13)(14,20,22,16)(19,23,24,21)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,13,15,17)(10,11,12,18)(14,21,22,23)(16,19,20,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(9,14,15,22)(10,16,12,20)(11,19,18,24)(13,21,17,23)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,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, 7908165);