#8094740

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

4405 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001936);