#7674166

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

4382 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,e^2=gi,f^2=g,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, 7581362);