#8926530

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	#58430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3853 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18,23,28)(19,27,30,24)(20,25,31,32)(21,26,29,22)(34,38)(36,43)(39,44)(42,47);(2,7)(3,10)(8,13)(11,16)(17,19,23,30)(18,24,28,27)(20,29,31,21)(22,25,26,32)(35,41)(39,44)(42,47)(46,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,16)(17,20,23,31)(18,25,28,32)(19,29,30,21)(22,24,26,27)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21)(18,26)(19,20)(22,28)(23,29)(24,32)(25,27)(30,31);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,27)(20,29)(21,31)(22,25)(23,30)(24,28)(26,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,22,23,26)(18,21,28,29)(19,25,30,32)(20,27,31,24)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833726);