#9910968

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

4222 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 9818164);