#9040411

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947607);