#8757177

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664373);