#8893422

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	#48552	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=g,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, 8800618);