#8496810

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	#53073	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404006);