#8898915

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,5)(4,9)(6,15)(7,8)(10,14)(11,12)(13,16)(17,18)(19,26)(20,25)(21,22)(23,24)(27,32)(28,29)(30,31)(33,34,39,37)(35,42,40,36)(38,45,46,41)(43,44,47,48);(1,3,7,6)(2,5,8,15)(4,10,13,12)(9,14,16,11)(18,24)(21,27)(25,26)(28,30)(34,37)(35,40)(41,45)(43,47);(2,8)(3,6)(9,16)(10,12)(17,19,23,20)(18,25,24,26)(21,28,27,30)(22,31,32,29)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,4)(2,9)(3,10)(5,14)(6,12)(7,13)(8,16)(11,15)(17,20,23,19)(18,26,24,25)(21,28,27,30)(22,31,32,29)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,5)(2,3)(4,11)(6,8)(7,15)(9,12)(10,16)(13,14)(17,21,23,27)(18,22,24,32)(19,28,20,30)(25,31,26,29)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,6,7,3)(2,5,8,15)(4,12,13,10)(9,14,16,11)(17,22)(18,27)(19,29)(20,31)(21,24)(23,32)(25,28)(26,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=g,e^2=g,f^2=gh,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, 8806111);