#8745950

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	#53778	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=hi,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, 8653146);