#8908351

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	#51630	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,29,32,31)(33,34)(35,41)(36,46)(37,43)(38,42)(39,40)(44,48)(45,47);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29)(33,35,40,37)(34,41,39,43)(36,45,38,44)(42,48,46,47);(3,11)(4,13)(7,14)(8,16)(17,18,22,26)(19,23,25,21)(20,24,30,28)(27,31,32,29)(33,36,40,38)(34,42,39,46)(35,44,37,45)(41,47,43,48);(1,4)(2,8)(3,12)(5,7)(6,13)(9,11)(10,16)(14,15)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32)(33,37,40,35)(34,43,39,41)(36,44,38,45)(42,47,46,48);(1,5)(2,9)(3,8)(4,14)(6,15)(7,13)(10,12)(11,16)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,29,23,31)(33,38,40,36)(34,42,39,46)(35,45,37,44)(41,47,43,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)(33,39,40,34)(35,43,37,41)(36,42,38,46)(44,47,45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8815547);