#8107633

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

3940 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)(18,21)(19,20)(26,29)(27,28)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(1,3,7,9)(2,8,4,5)(6,11,14,16)(10,15,12,13)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(2,4)(3,9)(10,12)(11,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,4,7,2)(3,8,9,5)(6,12,14,10)(11,15,16,13)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,5,7,8)(2,9,4,3)(6,13,14,15)(10,16,12,11)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,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, 8014829);