#8928747

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	#56441	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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)(4,9)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34)(35,39)(36,42)(37,45)(38,41)(40,46)(43,48)(44,47);(2,7)(4,14)(8,16)(10,12)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(3,11)(5,15)(8,16)(10,12)(19,27)(20,28)(22,30)(24,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835943);