#7736692

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	#59696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4445 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)(9,10,14,17)(11,15,18,12)(13,16,22,24)(19,23,21,20)(25,26,32,31)(27,28,34,30)(29,33,39,38)(35,36,40,37);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,20)(19,22)(23,24)(25,27)(26,30)(28,31)(29,35)(32,34)(33,37)(36,38)(39,40);(11,18)(12,15)(19,21)(20,23)(25,28,32,30)(26,27,31,34)(29,37,39,36)(33,40,38,35);(9,12,14,15)(10,11,17,18)(13,20,22,23)(16,19,24,21)(25,29,32,39)(26,33,31,38)(27,35,34,40)(28,36,30,37);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,24)(21,22)(25,30,32,28)(26,34,31,27)(29,37,39,36)(33,40,38,35);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,31,32,26)(27,28,34,30)(29,38,39,33)(35,36,40,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7643888);