#8994225

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	#35811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901421);