#8873639

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	#53870	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 8780835);