#8295571

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

4074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202767);