#8424544

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	#39382	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=gh,e^2=h,f^2=g,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, 8331740);