#8746292

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	#42629	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,23,25)(19,27,21,24)(20,26,30,31)(22,28,32,29)(33,34,39,37)(35,43,46,40)(36,41,45,47)(38,42,44,48);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,23,21)(18,24,25,27)(20,29,30,28)(22,26,32,31)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(3,12)(7,8)(13,16)(14,15)(18,25)(20,30)(24,27)(28,29)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32)(33,36)(34,41)(35,44)(37,47)(38,46)(39,45)(40,42)(43,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,21,23,19)(18,27,25,24)(20,29,30,28)(22,26,32,31)(33,37,39,34)(35,43,46,40)(36,47,45,41)(38,42,44,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,22,23,32)(18,28,25,29)(19,26,21,31)(20,27,30,24)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653488);