#7729017

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	#38325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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)(11,20)(12,17)(13,16)(14,18)(15,19)(21,24)(22,23)(25,26,29,33)(27,30,35,31)(28,32,37,40)(34,38,36,39);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,24,17,22)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(11,13)(16,20)(21,23)(22,24)(25,27,29,35)(26,31,33,30)(28,34,37,36)(32,39,40,38);(9,12,15,18)(10,17,19,14)(11,21,13,23)(16,22,20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,37)(39,40);(9,13,15,11)(10,16,19,20)(12,23,18,21)(14,24,17,22)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,14,15,17)(10,18,19,12)(11,22,13,24)(16,21,20,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=h,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, 7636213);