#7711428

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	#39006	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,24)(26,32)(27,30)(29,37)(36,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,26)(27,28)(29,34)(30,33)(31,32)(35,36)(37,39)(38,40);(10,16)(12,15)(17,23)(20,22)(25,27,31,30)(26,33,32,28)(29,38,37,35)(34,36,39,40);(9,12,14,15)(10,11,16,19)(13,20,21,22)(17,18,23,24)(25,28,31,33)(26,27,32,30)(29,36,37,40)(34,35,39,38);(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,29,31,37)(26,34,32,39)(27,35,30,38)(28,36,33,40);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,17,16,23)(11,18,19,24)(12,20,15,22)(25,30,31,27)(26,33,32,28)(29,38,37,35)(34,40,39,36)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=h,e^2=i,f^2=g,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, 7618624);