#8731089

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18,22,26)(19,23,27,24)(20,21,25,31)(28,29,32,30)(33,34,39,41)(35,40,44,47)(36,45,42,37)(38,43,46,48);(2,7)(3,11)(9,12)(10,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31)(33,35,39,44)(34,40,41,47)(36,46,42,38)(37,43,45,48);(2,7)(3,11)(9,12)(10,16)(19,27)(23,24)(28,32)(29,30)(34,41)(35,44)(36,42)(43,48);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,19)(18,24)(20,28)(21,30)(22,27)(23,26)(25,32)(29,31)(33,36,39,42)(34,37,41,45)(35,38,44,46)(40,48,47,43);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,20)(18,21)(19,28)(22,25)(23,29)(24,30)(26,31)(27,32)(33,37)(34,42)(35,43)(36,41)(38,40)(39,45)(44,48)(46,47);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,21)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30)(33,38)(34,43)(35,36)(37,47)(39,46)(40,45)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,e^2=g,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, 8638285);