#8746946

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=g,f^2=gi,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, 8654142);