#8417211

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	#44833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,48)(45,47);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(18,24)(19,20)(26,29)(27,28)(34,39)(40,43)(41,42)(47,48);(4,12)(7,14)(10,16)(13,15)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,35,38,46)(34,40,39,43)(36,44,37,45)(41,47,42,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,36,38,37)(34,41,39,42)(35,44,46,45)(40,47,43,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37,38,36)(34,42,39,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=i,e^2=i,f^2=g,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, 8324407);