#7712127

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	#36361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4461 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)(9,10,14,16)(11,15,18,13)(12,17,20,23)(19,22,24,21)(25,26,30,28)(27,32,35,31)(29,36,37,33)(34,38,40,39);(1,2,4,5)(3,6,7,8)(9,11,14,18)(10,15,16,13)(12,19,20,24)(17,22,23,21)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(10,16)(11,18)(12,20)(21,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(9,13,14,15)(10,18,16,11)(12,21,20,22)(17,24,23,19)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,e^2=g,f^2=i,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, 7619323);