#8824410

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 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)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,29,31)(35,41)(39,42)(43,47)(46,48);(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,19,22,24)(18,23,26,20)(21,29,30,27)(25,28,32,31)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,18,22,26)(19,23,24,20)(21,25,30,32)(27,31,29,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,29,32,27)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,22,30)(18,25,26,32)(19,27,24,29)(20,28,23,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731606);