#8445168

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

3935 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352364);