#8445965

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	#59685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,e^2=i,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, 8353161);