#8393690

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	#46026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(3,11,13,8)(4,12,14,9)(5,10,16,15)(18,23)(19,25)(26,30)(27,32);(2,7)(4,14)(8,11)(10,15)(17,18)(19,20)(21,26)(22,23)(24,25)(27,28)(29,30)(31,32)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(2,7)(5,16)(8,11)(9,12)(21,29)(26,30)(27,32)(28,31)(34,39)(37,45)(40,41)(43,48);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,12,16,9)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,12)(8,16)(13,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,5,6,16)(2,10,7,15)(3,12,13,9)(4,11,14,8)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=gh,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, 8300886);