#8110522

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	#45632	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(5,13)(7,10)(11,15)(21,29)(24,28)(26,32)(30,31)(33,34)(35,43)(36,39)(37,45)(38,46)(40,44)(41,47)(42,48);(2,4)(3,9)(8,12)(11,15)(18,20)(23,25)(24,28)(30,31)(34,36)(37,47)(40,43)(42,46);(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,18,22,20)(19,23,27,25)(21,24,29,28)(26,30,32,31)(33,35,39,44)(34,40,36,43)(37,42,47,46)(38,41,48,45);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,36,39,34)(35,43,44,40)(37,45,47,41)(38,46,48,42);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,37)(34,41)(35,42)(36,45)(38,43)(39,47)(40,48)(44,46);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,42)(35,41)(36,46)(37,43)(39,48)(40,47)(44,45)

Polycyclic presentation:

2^9|c^2=gi,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017718);