#8097190

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	#52690	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004386);