#8656194

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8563390);