#7729018

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	#49887	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4301 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=g,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, 7636214);