#9945612

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3808 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852808);