#8425660

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	#52330	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332856);