#8372151

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8279347);