#8107638

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(17,18,23,22)(19,26,24,20)(21,29,30,25)(27,28,31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,9)(2,8,4,5)(6,11,14,16)(10,15,12,13)(17,19,23,24)(18,20,22,26)(21,27,30,31)(25,28,29,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,4)(3,9)(10,12)(11,16)(17,20,23,26)(18,24,22,19)(21,28,30,32)(25,31,29,27)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,7,2)(3,8,9,5)(6,12,14,10)(11,15,16,13)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,7,8)(2,9,4,3)(6,13,14,15)(10,16,12,11)(17,21,23,30)(18,25,22,29)(19,27,24,31)(20,28,26,32);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22,23,18)(19,26,24,20)(21,29,30,25)(27,32,31,28)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,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, 8014834);