#8479567

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386763);