#10325412

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4237 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=h,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, 10232608);