#8492877

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

3920 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8400073);