#8258031

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	#42476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165227);