#8810318

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	#50285	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717514);