#8688395

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

3826 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)(7,15)(11,16)(13,14)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,2)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31)(34,38)(35,42)(40,44)(43,48);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4,6,12)(2,7,10,15)(3,11,9,16)(5,14,8,13)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,21,22,18)(19,25,27,23)(20,24,28,29)(26,30,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595591);