#8931473

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	#35405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8838669);