#8109902

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	#36944	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=ghi,e^2=gh,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, 8017098);