#9053993

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	#52693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(18,22)(23,28)(24,29)(30,32)(34,38)(35,41)(36,44)(45,48);(2,7)(4,11)(8,14)(12,16)(20,27)(24,29)(25,31)(30,32)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(3,9)(4,11)(5,13)(8,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,3,6,9)(2,4,7,11)(5,10,13,15)(8,12,14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,44)(41,47)(42,48);(1,4)(2,3)(5,12)(6,11)(7,9)(8,10)(13,16)(14,15)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,12,11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)

Polycyclic presentation:

2^9|b^2=i,d^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, 8961189);