#9053972

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	#57201	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,e^2=i,f^2=g,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, 8961168);