#8107635

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014831);