#8924646

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

4062 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)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18,23,25)(19,24,29,32)(20,26,30,31)(21,27,28,22);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,19,23,29)(18,24,25,32)(20,28,30,21)(22,31,27,26)(33,34,39,42)(35,43,37,40)(36,38,41,48)(44,47,46,45);(2,4)(5,13)(7,8)(10,16)(18,25)(19,29)(20,30)(22,27)(33,35,39,37)(34,40,42,43)(36,46,41,44)(38,47,48,45);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32)(33,36,39,41)(34,38,42,48)(35,44,37,46)(40,45,43,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21,23,28)(18,27,25,22)(19,30,29,20)(24,31,32,26)(33,37,39,35)(34,40,42,43)(36,44,41,46)(38,47,48,45);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,22,23,27)(18,28,25,21)(19,31,29,26)(20,24,30,32)(33,38)(34,41)(35,45)(36,42)(37,47)(39,48)(40,46)(43,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831842);