#8932354

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3849 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^2=gi,f^2=h,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, 8839550);