#8107571

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=i,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, 8014767);