#8932759

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(3,8)(4,13)(5,12)(6,7)(9,14)(10,15)(11,16)(19,25)(20,28)(23,26)(24,29)(34,40)(37,41)(42,47)(45,48);(2,7)(3,11)(9,13)(10,12)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,34,39,40)(35,41,36,37)(38,42,46,47)(43,48,44,45);(2,7)(5,15)(8,16)(9,13)(18,22)(23,26)(24,29)(31,32)(33,35,39,36)(34,41,40,37)(38,44,46,43)(42,45,47,48);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36,39,35)(34,41,40,37)(38,44,46,43)(42,48,47,45);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37,39,41)(34,36,40,35)(38,45,46,48)(42,44,47,43);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839955);