#8001017

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

4493 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 7908213);