#8734493

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	#55039	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(34,39)(35,42)(41,44)(43,48);(3,11)(4,12)(7,16)(8,13)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641689);