#7727011

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	#50594	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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)(9,10,15,18)(11,19,13,16)(12,14,17,24)(20,21,22,23)(25,26,31,33)(27,32,30,28)(29,37,39,34)(35,36,38,40);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,18,19)(12,20,17,22)(14,23,24,21)(25,27,31,30)(26,32,33,28)(29,35,39,38)(34,40,37,36);(11,13)(16,19)(20,22)(21,23)(26,33)(28,32)(34,37)(36,40);(1,3,4,7)(2,6,5,8)(9,12,15,17)(10,14,18,24)(11,20,13,22)(16,23,19,21)(25,28)(26,27)(29,36)(30,33)(31,32)(34,35)(37,38)(39,40);(9,13,15,11)(10,16,18,19)(12,22,17,20)(14,23,24,21)(25,29,31,39)(26,34,33,37)(27,35,30,38)(28,36,32,40);(9,14)(10,17)(11,21)(12,18)(13,23)(15,24)(16,20)(19,22)(25,30,31,27)(26,28,33,32)(29,38,39,35)(34,36,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7634207);