#8011964

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	#39958	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

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, 7919160);