#8446076

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8353272);