#8772938

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680134);