#9794279

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	#38702	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,38,42)(35,39,44,45)(36,40,46,48)(37,41,47,43);(2,6)(7,9)(8,12)(14,15)(18,24)(19,21)(26,30)(27,29)(33,35,38,44)(34,39,42,45)(36,41,46,43)(37,40,47,48);(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,35)(34,39)(36,43)(37,40)(38,44)(41,46)(42,45)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,36)(34,40)(35,43)(37,39)(38,46)(41,44)(42,48)(45,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37)(34,41)(35,40)(36,45)(38,47)(39,46)(42,43)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=h,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, 9701475);