#8179752

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	#45515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(18,20)(19,25)(24,28)(27,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

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

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, 8086948);