#9685578

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9592774);