#8461194

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	#45645	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=ghi,f^2=i,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, 8368390);