#9910701

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	#53206	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=g,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, 9817897);