#9481433

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	#55031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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)(3,8)(4,14)(5,11)(6,15)(7,9)(10,13)(12,16)(19,25)(23,30)(26,28)(31,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,12)(2,8,9,16)(4,5,13,15)(6,14,11,10)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,9)(3,12)(4,13)(6,11)(18,22)(23,30)(24,27)(31,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,10)(3,5)(6,8)(7,13)(9,14)(11,16)(12,15)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5)(2,11)(3,13)(4,12)(6,9)(7,15)(8,10)(14,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47);(1,6,7,11)(2,5,9,15)(3,14,12,10)(4,8,13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,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, 9388629);