#8901754

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

4053 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808950);