#8445934

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	#53530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=i,f^2=g,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, 8353130);