#8745961

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3861 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,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, 8653157);