#8445903

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	#36777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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:

(1,2,8,11)(3,9,6,7)(4,14,5,10)(12,15,13,16)(17,18)(19,28)(20,25)(21,29)(22,30)(23,26)(24,27)(31,32)(34,39)(35,41)(36,44)(43,48);(1,3,8,6)(2,9,11,7)(4,13,5,12)(10,15,14,16)(17,19,24,26)(18,23,27,28)(20,29,31,22)(21,25,30,32)(33,34,38,39)(35,40,41,46)(36,45,44,37)(42,48,47,43);(1,4,8,5)(2,10,11,14)(3,12,6,13)(7,15,9,16)(17,20,24,31)(18,25,27,32)(19,29,26,22)(21,23,30,28)(33,35)(34,40)(36,42)(37,43)(38,41)(39,46)(44,47)(45,48);(1,5,8,4)(2,10,11,14)(3,13,6,12)(7,15,9,16)(17,21)(18,22)(19,25)(20,23)(24,30)(26,32)(27,29)(28,31)(33,36,38,44)(34,37,39,45)(35,42,41,47)(40,43,46,48);(1,6,8,3)(2,7,11,9)(4,13,5,12)(10,15,14,16)(17,22)(18,21)(19,20)(23,25)(24,29)(26,31)(27,30)(28,32);(1,7)(2,3)(4,15)(5,16)(6,11)(8,9)(10,12)(13,14)(17,23)(18,26)(19,27)(20,30)(21,31)(22,25)(24,28)(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=gi,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, 8353099);