#8917880

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	#50711	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(33,34,38,43)(35,40,45,39)(36,41,47,44)(37,46,48,42);(2,7)(4,8)(9,13)(12,15)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(33,34,38,43)(35,40,45,39)(36,44,47,41)(37,42,48,46);(2,7)(3,10)(5,14)(12,15)(18,23)(19,26)(21,30)(27,29)(33,35,38,45)(34,39,43,40)(36,42,47,46)(37,41,48,44);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,43)(41,48)(42,47);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36,38,47)(34,41,43,44)(35,42,45,46)(37,39,48,40);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37)(34,42)(35,44)(36,39)(38,48)(40,47)(41,45)(43,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=gi,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, 8825076);