#8471038

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=ghi,e^2=hi,f^2=gi,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, 8378234);