#8679896

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8587092);