#8922733

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	#51395	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=ghi,e^2=i,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, 8829929);