#9040894

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	#51941	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4138 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^2=g,f^2=gh,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, 8948090);