#8446021

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	#42052	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353217);