#8425115

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	#59824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4312 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:

(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,24,23)(26,31)(28,34)(32,38)(36,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,21)(10,15,18,19)(12,13,20,24)(16,17,22,23)(27,35)(32,38)(36,40)(37,39);(1,3,5,6)(2,4,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(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)(11,19)(12,16)(13,23)(14,18)(15,21)(17,24)(20,22)(25,27,30,35)(26,32,31,38)(28,36,34,40)(29,37,33,39);(9,12,14,20)(10,16,18,22)(11,13,21,24)(15,17,19,23)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,20)(12,21)(14,24)(15,22)(16,19)(18,23)(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=i,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, 8332311);