#9912897

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	#2671	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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,34,37,40)(35,38,41,45)(36,39,44,48)(42,46,43,47);(3,9)(7,13)(10,15)(14,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(35,41)(38,45)(42,43)(46,47);(2,6)(7,13)(8,11)(14,16)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,44)(47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,35)(34,38)(36,43)(37,41)(39,47)(40,45)(42,44)(46,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,48)(41,43)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=h,f^2=h,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, 9820093);