#8745843

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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)(3,12)(4,9)(5,8)(6,10)(7,11)(13,14)(15,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(1,3,7,5)(2,8,11,12)(4,13,10,15)(6,14,9,16)(19,25)(22,28)(26,31)(29,32)(34,40)(35,42)(41,44)(43,48);(1,4,7,10)(2,9,11,6)(3,13,5,15)(8,16,12,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,4)(2,9)(3,13)(5,15)(6,11)(7,10)(8,16)(12,14)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5,7,3)(2,8,11,12)(4,15,10,13)(6,14,9,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,6,7,9)(2,10,11,4)(3,14,5,16)(8,15,12,13)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=hi,e^2=g,f^2=gh,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, 8653039);