#8260689

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	#38516	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=gh,e^2=hi,f^2=ghi,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, 8167885);