#8944436

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	#58100	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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)(3,4)(5,13)(6,7)(8,14)(9,10)(11,16)(12,15)(18,19)(21,29)(23,25)(27,32)(33,34,38,41)(35,39,45,48)(36,40,46,37)(42,43,44,47);(1,3)(2,4)(5,11)(6,9)(7,10)(8,12)(13,16)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46);(2,7)(3,9)(5,14)(12,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(33,36,38,46)(34,40,41,37)(35,42,45,44)(39,43,48,47);(1,4)(2,3)(5,12)(6,10)(7,9)(8,11)(13,15)(14,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,37)(34,36)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,2)(3,10)(4,9)(5,8)(6,7)(11,16)(12,15)(13,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,5,6,14)(2,8,7,13)(3,11,9,15)(4,12,10,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,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,e^c=egi,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, 8851632);