#9053950

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4144 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961146);