#8107593

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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:

(1,2,7,12)(3,8,14,15)(4,11,6,9)(5,10,16,13)(18,21)(19,27)(20,30)(26,32)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(1,3,7,14)(2,8,12,15)(4,10,6,13)(5,11,16,9)(17,18,23,21)(19,28,27,24)(20,31,30,25)(22,26,29,32)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(1,4,7,6)(2,9,12,11)(3,13,14,10)(5,8,16,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,27)(25,32)(26,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,5,7,16)(2,10,12,13)(3,9,14,11)(4,15,6,8)(17,20,23,30)(18,25,21,31)(19,29,27,22)(24,32,28,26)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,6,7,4)(2,11,12,9)(3,13,14,10)(5,8,16,15)(17,21,23,18)(19,28,27,24)(20,31,30,25)(22,32,29,26);(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,16)(14,15)(17,22,23,29)(18,26,21,32)(19,30,27,20)(24,31,28,25)(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=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, 8014789);