#9712452

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	#48187	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18,22,26)(19,25,29,23)(20,21,30,24)(27,31,32,28)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,48,44,42);(1,3,7,11)(2,5,8,16)(4,12,14,6)(9,15,13,10)(17,19,22,29)(18,23,26,25)(20,27,30,32)(21,28,24,31)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,42,46,48);(2,8)(3,11)(4,14)(10,15)(17,20,22,30)(18,24,26,21)(19,27,29,32)(23,31,25,28)(33,36,39,47)(34,41,37,46)(35,38,45,44)(40,48,43,42);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,20)(18,21)(19,27)(22,30)(23,28)(24,26)(25,31)(29,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,42)(43,48);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,19)(18,25)(20,27)(21,31)(22,29)(23,26)(24,28)(30,32)(33,37,39,34)(35,40,45,43)(36,46,47,41)(38,48,44,42);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,21)(18,20)(19,28)(22,24)(23,27)(25,32)(26,30)(29,31)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,46,43,41)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619648);