#8093712

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

4432 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,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, 8000908);