#9910892

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	#53612	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818088);