#8757451

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	#56872	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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)(4,8,12,13)(11,15,16,14)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,38,39)(35,41,44,40)(36,37,45,46)(42,47,48,43);(2,6)(4,12)(7,9)(11,16)(21,29)(25,28)(27,30)(31,32)(34,39)(36,45)(40,41)(42,48);(3,10)(7,9)(11,16)(14,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,46)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,35,38,44)(34,40,39,41)(36,42,45,48)(37,43,46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,47)(44,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,37)(34,36)(35,43)(38,46)(39,45)(40,42)(41,48)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664647);