#8127477

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	#36923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034673);