#8093707

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
prime radical	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4264 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)(11,14)(12,17)(13,23)(15,18)(16,21)(19,24)(20,22)(26,27)(28,35)(32,34)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,20)(14,18)(15,21)(17,22)(23,24)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(2,5)(6,8)(9,12,15,13)(10,17,18,23)(11,19,21,20)(14,24,16,22)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(9,13,15,12)(10,17,18,23)(11,20,21,19)(14,24,16,22)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,16)(10,11,18,21)(12,22,13,24)(17,20,23,19)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,20)(14,18)(15,21)(17,22)(23,24)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^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, 8000903);