#8197203

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	#53572	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(34,35)(36,43)(40,42)(44,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,18)(19,21)(20,25)(22,23)(24,27)(26,28)(29,30)(31,32)(33,34,38,35)(36,39,43,41)(37,40,45,42)(44,46,48,47);(2,6)(7,13)(8,11)(14,15)(18,23)(21,24)(25,30)(28,31)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,20,22,29)(18,25,23,30)(19,26,27,32)(21,28,24,31)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,21,22,24)(18,19,23,27)(20,28,29,31)(25,26,30,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8104399);