#8931092

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,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, 8838288);