#8445311

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352507);