#8901478

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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=ghi,e^2=ghi,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, 8808674);