#8926249

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	#36921	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8833445);