#8739417

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	#44321	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,23,26)(19,27,21,24)(20,22,25,32)(28,31,30,29)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(2,7)(3,8)(9,16)(10,13)(17,19,23,21)(18,24,26,27)(20,28,25,30)(22,29,32,31)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(2,7)(3,8)(9,16)(10,13)(17,20,23,25)(18,22,26,32)(19,28,21,30)(24,29,27,31)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,32)(27,31)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21,23,19)(18,24,26,27)(20,30,25,28)(22,29,32,31)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,22)(18,25)(19,29)(20,26)(21,31)(23,32)(24,30)(27,28)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,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, 8646613);