#8268581

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

4164 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:

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

Polycyclic presentation:

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

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, 8175777);