#8762006

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34,38,39)(35,41,44,40)(36,37,45,46)(42,47,48,43);(2,6)(4,13)(7,9)(11,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(34,39)(36,45)(40,41)(42,48);(2,6)(3,10)(8,12)(11,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,46)(47,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(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)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,47)(44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(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=gh,d^2=g,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, 8669202);