#9040952

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

3971 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=gh,f^2=h,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, 8948148);