#7673617

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18)(19,24)(20,25)(21,33)(22,23)(26,34)(27,35)(28,36)(29,43)(30,32)(31,44)(37,40)(38,45)(39,41)(42,48)(46,47);(9,10,12,13)(11,14,15,16)(18,23)(19,27)(25,30)(26,33)(28,39)(29,40)(38,44)(42,47);(2,5)(6,8)(17,18)(19,24)(20,30)(21,26)(22,23)(25,32)(27,35)(28,41)(29,37)(31,44)(33,34)(36,39)(38,45)(40,43)(42,48)(46,47);(1,2)(3,6)(4,5)(7,8)(17,19,22,27)(18,24,23,35)(20,28,32,39)(21,29,34,40)(25,36,30,41)(26,37,33,43)(31,42,45,47)(38,46,44,48);(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,25,23,30)(19,28,27,39)(21,31,34,45)(24,36,35,41)(26,38,33,44)(29,42,40,47)(37,46,43,48);(1,3,4,7)(2,6,5,8)(17,21)(18,26)(19,29)(20,31)(22,34)(23,33)(24,37)(25,38)(27,40)(28,42)(30,44)(32,45)(35,43)(36,46)(39,47)(41,48)

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7580813);