#9481453

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	#53298	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3780 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)(3,7,10,11)(5,8,13,12)(9,14,15,16)(17,18,23,21)(19,22,28,24)(20,26,30,25)(27,29,32,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,4)(7,11)(8,12)(14,16)(17,19,23,28)(18,24,21,22)(20,27,30,32)(25,29,26,31)(34,38)(39,41)(40,45)(46,47);(5,13)(8,12)(9,15)(14,16)(17,20,23,30)(18,25,21,26)(19,27,28,32)(22,31,24,29)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,6,10)(2,7,4,11)(5,9,13,15)(8,14,12,16)(17,20)(18,26)(19,27)(21,25)(22,29)(23,30)(24,31)(28,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,21,23,18)(19,22,28,24)(20,26,30,25)(27,31,32,29)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,5)(2,8)(3,9)(4,12)(6,13)(7,14)(10,15)(11,16)(17,22)(18,19)(20,29)(21,28)(23,24)(25,32)(26,27)(30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,e^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, 9388649);