#8898886

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

4002 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806082);