#8843925

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	#45506	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8751121);