#8818012

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(18,23)(20,25)(21,30)(26,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,7)(3,9)(8,11)(10,12)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4,5,13)(2,8,7,11)(3,10,9,12)(6,14,15,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725208);