#8873600

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	#58242	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

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

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, 8780796);