#8917881

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	#51085	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825077);