#8455693

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	#45647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,22)(23,24)(26,31)(28,32)(34,37)(36,39);(2,3)(6,7)(9,11,15,13)(10,16,20,17)(12,23,19,21)(14,22,18,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2,5,3)(4,6,8,7)(9,11,15,13)(10,17,20,16)(12,21,19,23)(14,22,18,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,3,5,2)(4,7,8,6)(9,12,15,19)(10,18,20,14)(11,21,13,23)(16,22,17,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,4,5,8)(2,6,3,7)(9,13,15,11)(10,16,20,17)(12,23,19,21)(14,24,18,22)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,14,15,18)(10,19,20,12)(11,22,13,24)(16,21,17,23)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362889);