#8192425

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

4496 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8099621);