#8445918

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=gi,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, 8353114);