#8112457

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	#56771	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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)(35,41)(36,40)(37,45)(38,48)(39,42)(43,46)(44,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(20,28)(25,30)(27,29)(31,32)(33,35,39,45)(34,37,42,41)(36,43,38,44)(40,47,48,46);(2,6)(7,9)(8,13)(14,15)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,36,39,38)(34,40,42,48)(35,43,45,44)(37,47,41,46);(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)(33,37,39,41)(34,35,42,45)(36,46,38,47)(40,44,48,43);(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)(33,35)(34,41)(36,43)(37,42)(38,44)(39,45)(40,46)(47,48);(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)(33,38,39,36)(34,40,42,48)(35,44,45,43)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gh,e^2=h,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, 8019653);