#8534602

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

4272 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=h,e^2=gi,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, 8441798);