#8751297

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gi,e^2=gh,f^2=gi,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, 8658493);