#8389279

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	#56313	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8296475);