#8445985

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

4007 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)(3,11)(4,15)(5,16)(6,10)(7,14)(8,13)(9,12)(17,18,23,26)(19,27,20,24)(21,22,25,32)(28,31,30,29)(34,40)(35,43)(36,45)(42,47);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,11)(15,16)(19,20)(24,27)(28,30)(29,31)(33,34,39,40)(35,38,43,48)(36,46,45,41)(37,42,44,47);(17,19,23,20)(18,24,26,27)(21,28,25,30)(22,29,32,31)(33,35,39,43)(34,38,40,48)(36,44,45,37)(41,42,46,47);(1,4,6,13)(2,8,10,15)(3,12,14,5)(7,9,11,16)(17,20,23,19)(18,24,26,27)(21,30,25,28)(22,29,32,31)(33,36)(34,41)(35,44)(37,43)(38,42)(39,45)(40,46)(47,48);(1,5,6,12)(2,9,10,16)(3,13,14,4)(7,8,11,15)(17,21,23,25)(18,22,26,32)(19,28,20,30)(24,29,27,31)(33,37,39,44)(34,42,40,47)(35,36,43,45)(38,46,48,41);(1,2)(3,7)(4,15)(5,9)(6,10)(8,13)(11,14)(12,16)(17,22)(18,25)(19,29)(20,31)(21,26)(23,32)(24,30)(27,28)(33,38)(34,43)(35,40)(36,47)(37,46)(39,48)(41,44)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,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, 8353181);