#8343529

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

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

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, 8250725);