#8107577

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

3990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,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, 8014773);