#8756141

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

3876 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:

(3,9)(7,10)(11,15)(14,16)(17,18,23,28)(19,26,21,24)(20,27,22,25)(29,31,30,32)(35,37)(39,41)(43,45)(47,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,23,21)(18,24,28,26)(20,29,22,30)(25,31,27,32)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(2,6)(7,10)(8,12)(14,16)(17,20,23,22)(18,25,28,27)(19,29,21,30)(24,31,26,32)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,26,28,24)(20,29,22,30)(25,31,27,32);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,31)(30,32)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,20)(18,27,28,25)(19,29,21,30)(24,31,26,32)(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=h,c^2=h,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, 8663337);