#9054423

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	#41967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(3,8,4,13)(5,10,15,14)(6,12,16,11)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(1,3,7,4)(2,8,9,13)(5,12,15,11)(6,10,16,14)(17,19,22,24)(18,20,23,26)(21,27,30,31)(25,28,29,32)(37,45)(40,44)(43,47)(46,48);(2,9)(3,4)(5,15)(6,16)(18,23)(20,26)(21,30)(27,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,4,7,3)(2,8,9,13)(5,12,15,11)(6,14,16,10)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,5)(2,10)(3,12)(4,11)(6,13)(7,15)(8,16)(9,14)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,6,7,16)(2,11,9,12)(3,10,4,14)(5,13,15,8)(17,21,22,30)(18,25,23,29)(19,27,24,31)(20,28,26,32)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961619);