#8276809

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

4224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8184005);