#8908181

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8815377);