#9058457

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	#55012	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8965653);