#9481523

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	#54201	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gi,e^2=h,f^2=g,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, 9388719);