#8534536

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

3960 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(2,8)(4,15)(7,11)(10,12)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441732);