#8898022

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

4655 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

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, 8805218);