#9240970

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

3710 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9148166);