#9863280

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770476);