#8417334

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	#52757	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324530);