#8581426

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3885 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8488622);