#8688366

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	#45604	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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)(17,18,22,26)(19,23,28,25)(20,29,21,24)(27,32,30,31)(33,34)(35,47)(36,42)(37,43)(38,44)(39,45)(40,46)(41,48);(1,2)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,32)(25,26)(29,31)(33,35)(34,41)(36,37)(38,45)(39,43)(40,48)(42,44)(46,47);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(17,20,22,21)(18,24,26,29)(19,27,28,30)(23,31,25,32)(33,36,40,45)(34,42,46,39)(35,37,48,38)(41,44,47,43);(1,4,6,12)(2,7,10,15)(3,11,9,16)(5,14,8,13)(17,18)(19,25)(20,24)(21,29)(22,26)(23,28)(27,32)(30,31)(33,37,40,38)(34,43,46,44)(35,36,48,45)(39,47,42,41);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,19)(18,25)(20,27)(21,30)(22,28)(23,26)(24,32)(29,31)(33,38,40,37)(34,44,46,43)(35,45,48,36)(39,41,42,47);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,21,22,20)(18,24,26,29)(19,27,28,30)(23,32,25,31)(33,39,40,42)(34,45,46,36)(35,43,48,44)(37,41,38,47)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=ghi,e^2=g,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, 8595562);