#8445912

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8353108);