#10361130

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	#54486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10268326);