#8174386

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	#40008	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081582);