#8924768

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3927 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)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(33,34,38,42)(35,41,45,39)(36,46,40,37)(43,48,47,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,18,23,26)(19,27,21,24)(20,22,25,32)(28,31,30,29)(33,35,38,45)(34,39,42,41)(36,47,40,43)(37,44,46,48);(1,2,5,8)(3,6,11,9)(4,12,13,7)(10,15,16,14)(17,19,23,21)(18,24,26,27)(20,30,25,28)(22,31,32,29)(33,36,38,40)(34,37,42,46)(35,43,45,47)(39,48,41,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,20,23,25)(18,22,26,32)(19,28,21,30)(24,29,27,31)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,21,23,19)(18,24,26,27)(20,28,25,30)(22,31,32,29)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,47)(46,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22)(18,25)(19,29)(20,26)(21,31)(23,32)(24,30)(27,28)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831964);