#8097064

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=hi,f^2=hi,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, 8004260);