#8678975

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,15)(14,16)(18,23)(19,20)(26,29)(27,28)(33,34,40,37)(35,41,39,43)(36,38,47,42)(44,48,46,45);(1,3)(2,9)(4,6)(5,14)(7,11)(8,15)(10,16)(12,13)(17,18,22,23)(19,24,20,25)(21,26,30,29)(27,31,28,32)(33,35,40,39)(34,41,37,43)(36,46,47,44)(38,45,42,48);(1,4,8,5)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,36,40,47)(34,42,37,38)(35,44,39,46)(41,45,43,48);(1,5,8,4)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,37,40,34)(35,41,39,43)(36,38,47,42)(44,45,46,48);(1,6,8,14)(2,11,12,16)(3,4,15,5)(7,13,10,9)(17,20,22,19)(18,25,23,24)(21,27,30,28)(26,31,29,32)(33,38)(34,36)(35,45)(37,47)(39,48)(40,42)(41,46)(43,44);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,9,14,13)(17,21,22,30)(18,26,23,29)(19,27,20,28)(24,31,25,32)(33,39,40,35)(34,43,37,41)(36,46,47,44)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8586171);