#8734819

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	#46469	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=g,e^2=h,f^2=g,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, 8642015);