#9910925

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 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)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(35,42)(37,39)(43,47)(44,46);(3,9)(6,13)(10,15)(14,16)(18,23)(24,25)(26,29)(31,32)(33,34)(35,37)(36,40)(38,41)(39,42)(43,46)(44,47)(45,48);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,35,38,42)(34,39,41,37)(36,43,45,47)(40,44,48,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,22,19)(18,25,23,24)(21,28,30,27)(26,32,29,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,46)(44,48)(45,47);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,36,38,45)(34,40,41,48)(35,43,42,47)(37,46,39,44);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21,22,30)(18,26,23,29)(19,27,20,28)(24,31,25,32)(33,37)(34,35)(36,44)(38,39)(40,47)(41,42)(43,48)(45,46)

Polycyclic presentation:

2^9|c^2=gh,d^2=g,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, 9818121);