#8463685

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

3996 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(35,43)(36,45)(39,46)(40,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(18,23)(19,26)(25,28)(27,32)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,45)(34,40,42,48)(35,44,43,37)(39,47,46,41);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37,38,44)(34,41,42,47)(35,45,43,36)(39,48,46,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=g,e^2=ghi,f^2=ghi,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, 8370881);