#8893618

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	#49661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8800814);