#8393614

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,11)(4,13)(5,10)(6,7)(8,14)(9,15)(12,16)(33,34,39,41)(35,37,43,36)(38,40,46,48)(42,45,47,44);(2,7)(4,15)(8,11)(10,12)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(33,35,39,43)(34,36,41,37)(38,42,46,47)(40,44,48,45);(2,7)(5,16)(8,11)(9,13)(18,23)(21,29)(24,25)(27,32)(33,36,39,37)(34,35,41,43)(38,45,46,44)(40,47,48,42);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,37,39,36)(34,35,41,43)(38,45,46,44)(40,42,48,47);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,35)(34,37)(36,41)(38,42)(39,43)(40,45)(44,48)(46,47);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,46)(34,40,41,48)(35,42,43,47)(36,44,37,45)

Polycyclic presentation:

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

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, 8300810);