#9873175

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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)(8,13)(9,16)(10,14)(12,15)(33,34)(35,46)(36,41)(37,47)(38,43)(39,44)(40,45)(42,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31)(33,35)(34,36)(37,43)(38,42)(39,48)(40,41)(44,47)(45,46);(2,6)(7,14)(8,11)(15,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,36,40,46)(34,41,45,35)(37,38,48,44)(39,47,43,42);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,37,40,48)(34,42,45,47)(35,43,41,39)(36,38,46,44);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,19)(18,25,20,24)(22,27,30,29)(26,31,28,32)(33,38)(34,43)(35,47)(36,48)(37,46)(39,45)(40,44)(41,42);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39,40,43)(34,44,45,38)(35,48,41,37)(36,47,46,42)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=ghi,e^2=i,f^2=g,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, 9780371);