#8425118

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4275 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332314);