#8930729

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	#36817	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837925);