#8395010

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

4111 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:

(1,2,6,7)(3,9,11,5)(4,12,15,8)(10,14,13,16)(33,34,39,41)(35,37,43,36)(38,40,46,48)(42,45,47,44);(1,3)(2,5)(4,13)(6,11)(7,9)(8,16)(10,15)(12,14)(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)(5,9)(8,12)(14,16)(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)(2,5)(4,13)(6,11)(7,9)(8,16)(10,15)(12,14)(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)(2,8,7,12)(3,10,11,13)(5,14,9,16)(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)(2,3)(4,14)(6,9)(7,11)(8,10)(12,13)(15,16)(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=g,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,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, 8302206);