#7711101

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	#45687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618297);