#8093618

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	#35676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000814);