#8096829

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

(1,2,6,10)(3,7,4,8)(5,9,13,16)(11,14,12,15)(17,18,23,27)(19,21,29,24)(20,30,28,25)(22,26,31,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,31)(21,27)(22,28)(23,29)(25,32)(26,30)(35,42)(39,46)(43,47)(44,48);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,20,23,28)(18,25,27,30)(19,22,29,31)(21,32,24,26)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,21)(18,19)(20,32)(22,25)(23,24)(26,28)(27,29)(30,31);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,19)(18,21)(20,22)(23,29)(24,27)(25,32)(26,30)(28,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,22)(18,26)(19,28)(20,29)(21,25)(23,31)(24,30)(27,32)(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=g,e^2=gh,f^2=gh,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, 8004025);