#8739400

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3811 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)(3,9)(4,8)(5,13)(6,10)(7,15)(11,14)(12,16)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,30)(31,32)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(1,3)(2,5)(4,11)(6,13)(7,14)(8,16)(9,10)(12,15)(17,19,23,21)(18,24,28,25)(20,31,27,29)(22,30,26,32)(34,40)(35,42)(41,44)(43,48);(1,4,6,15)(2,7,10,8)(3,11,13,12)(5,14,9,16)(17,19,23,21)(18,25,28,24)(20,29,27,31)(22,30,26,32)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,4)(2,8)(3,12)(5,14)(6,15)(7,10)(9,16)(11,13)(17,20,23,27)(18,26,28,22)(19,29,21,31)(24,30,25,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5)(2,3)(4,14)(6,9)(7,11)(8,12)(10,13)(15,16)(17,21,23,19)(18,24,28,25)(20,31,27,29)(22,32,26,30)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,3)(2,9)(4,11)(5,10)(6,13)(7,16)(8,14)(12,15)(17,22,23,26)(18,27,28,20)(19,30,21,32)(24,29,25,31)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=gh,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, 8646596);