#8551148

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	#40829	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458344);