#8528876

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

4186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,e^2=gh,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, 8436072);