#9685577

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	#2535	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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,7)(4,15)(8,11)(10,13)(17,19,23,21)(18,24,22,26)(20,27,30,29)(25,31,28,32)(35,42)(39,46)(43,47)(44,48);(2,7)(3,12)(9,14)(10,13)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,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, 9592773);