#8192421

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099617);