#10439964

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	#57012	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(4,11)(8,10)(12,16)(17,18,23,21)(19,26,20,24)(22,29,30,25)(27,31,28,32)(34,38)(36,43)(39,45)(41,44);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(34,38)(39,45)(40,46)(47,48);(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,46)(44,48);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,35,37,42)(34,39,38,45)(36,44,43,41)(40,48,46,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,46)(44,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,36)(34,40)(35,41)(37,43)(38,46)(39,47)(42,44)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10347160);