#8926555

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	#47010	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,e^2=g,f^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, 8833751);