#8931004

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

4431 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(2,5)(6,7)(10,15)(12,21)(16,23)(18,22)(29,37)(32,36)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(11,19)(13,24)(16,23)(18,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,20)(14,19)(15,23)(17,24)(18,21)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,23,24,16)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(9,13)(10,18)(11,20)(12,23)(14,24)(15,22)(16,21)(17,19)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,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, 8838200);