#9481521

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	#54723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3769 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 9388717);