#8745970

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3743 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)(3,10)(4,6)(5,9)(7,8)(11,12)(13,16)(14,15)(18,23)(20,26)(24,30)(28,32)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(2,8)(4,13)(9,10)(11,14)(19,27)(24,30)(28,32)(29,31)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(1,3,7,5)(2,9,8,10)(4,14,13,11)(6,12,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,4,7,13)(2,6,8,16)(3,11,5,14)(9,15,10,12)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,31)(26,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,5,7,3)(2,10,8,9)(4,14,13,11)(6,12,16,15)(17,20,22,26)(18,25,23,21)(19,28,27,32)(24,31,30,29);(1,6)(2,4)(3,12)(5,15)(7,16)(8,13)(9,14)(10,11)(17,21,22,25)(18,26,23,20)(19,29,27,31)(24,32,30,28)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653166);