#8746090

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	#42138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=hi,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, 8653286);