#8934296

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841492);