#8533550

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	#55266	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=g,e^2=gh,f^2=gh,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, 8440746);