#8112462

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3784 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8019658);