#8391328

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4387 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:

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

Polycyclic presentation:

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

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, 8298524);