#9040879

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	#35508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(6,7)(8,10)(11,14)(12,15)(13,16)(17,18,22,24)(19,23,26,20)(21,29,30,25)(27,28,32,31);(2,7)(3,10)(9,15)(11,13)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(2,7)(4,12)(5,8)(11,13)(18,24)(19,26)(21,30)(28,31)(34,38)(36,45)(39,41)(42,46);(1,3,6,10)(2,8,7,5)(4,13,12,11)(9,14,15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,45)(44,47);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,16)(10,13)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,5,6,8)(2,10,7,3)(4,14,12,16)(9,13,15,11)(17,21,22,30)(18,25,24,29)(19,27,26,32)(20,28,23,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948075);