#8505012

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

4473 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)(9,10,15,16)(11,17,12,21)(13,22,23,18)(14,19,24,20)(25,26,30,34)(27,33,37,31)(28,38,29,32)(35,39,36,40);(2,5)(6,7)(10,16)(13,23)(17,21)(19,20)(25,27,30,37)(26,31,34,33)(28,35,29,36)(32,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,21)(13,20,23,19)(14,18,24,22)(25,28,30,29)(26,32,34,38)(27,35,37,36)(31,39,33,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,17,16,21)(13,19,23,20)(14,18,24,22)(25,26)(27,33)(28,32)(29,38)(30,34)(31,37)(35,40)(36,39);(9,13)(10,18)(11,20)(12,19)(14,21)(15,23)(16,22)(17,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,34)(32,40)(38,39);(9,14,15,24)(10,19,16,20)(11,18,12,22)(13,21,23,17)(25,29,30,28)(26,32,34,38)(27,36,37,35)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8412208);