#9910920

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	#53390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(19,26)(21,23)(27,31)(28,30)(33,34,39,41)(35,42,36,37)(38,40,47,48)(43,46,44,45);(3,9)(6,13)(10,15)(14,16)(17,18)(19,21)(20,24)(22,25)(23,26)(27,30)(28,31)(29,32)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,48,46);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,22,26)(18,23,25,21)(20,27,29,31)(24,28,32,30)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,48,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,30)(28,32)(29,31);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,22,29)(18,24,25,32)(19,27,26,31)(21,30,23,28)(33,37)(34,35)(36,41)(38,46)(39,42)(40,44)(43,48)(45,47);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21)(18,19)(20,28)(22,23)(24,31)(25,26)(27,32)(29,30)(33,38,39,47)(34,40,41,48)(35,43,36,44)(37,45,42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=h,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, 9818116);