#8395014

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=hi,e^2=gh,f^2=h,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, 8302210);