#9039140

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	#55882	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(33,34,39,37)(35,40,38,46)(36,43,45,41)(42,48,44,47);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(18,23)(19,26)(24,29)(27,32)(33,35,39,38)(34,40,37,46)(36,42,45,44)(41,47,43,48);(2,6)(3,5)(9,12)(11,14)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,36)(34,41)(35,42)(37,43)(38,44)(39,45)(40,47)(46,48);(1,4,7,13)(2,9,6,12)(3,11,5,14)(8,15,10,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,5,7,3)(2,10,6,8)(4,11,13,14)(9,15,12,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,37,39,34)(35,40,38,46)(36,43,45,41)(42,47,44,48);(1,6,7,2)(3,10,5,8)(4,12,13,9)(11,16,14,15)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,35)(34,40,37,46)(36,44,45,42)(41,47,43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946336);