#9481411

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

3925 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388607);