#8688403

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	#42892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(17,18,23,22)(19,26,30,24)(20,25,29,31)(21,27,28,32)(33,34,39,41)(35,42,45,37)(36,46,38,40)(43,47,44,48);(1,2)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,26)(23,30)(25,27)(31,32)(33,35,39,45)(34,37,41,42)(36,43,38,44)(40,47,46,48);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(17,20,23,29)(18,25,22,31)(19,28,30,21)(24,27,26,32)(33,36,39,38)(34,40,41,46)(35,43,45,44)(37,47,42,48);(1,4,6,12)(2,7,10,15)(3,11,9,16)(5,14,8,13)(17,19)(18,26)(20,28)(21,29)(22,24)(23,30)(25,32)(27,31)(33,37)(34,35)(36,47)(38,48)(39,42)(40,43)(41,45)(44,46);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,21,23,28)(18,27,22,32)(19,29,30,20)(24,25,26,31)(33,34)(35,37)(36,40)(38,46)(39,41)(42,45)(43,47)(44,48);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,22,23,18)(19,26,30,24)(20,31,29,25)(21,27,28,32)(33,38,39,36)(34,40,41,46)(35,44,45,43)(37,47,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=gh,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, 8595599);