#8464275

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8371471);