#8754979

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	#51960	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,11,12,8)(4,14,15,9)(5,13,16,10)(19,21)(22,30)(24,26)(25,28)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(2,7)(5,16)(8,11)(9,14)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(34,40)(35,44)(37,47)(42,45);(2,7)(3,12)(9,14)(10,13)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,3,6,12)(2,8,7,11)(4,13,15,10)(5,9,16,14)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)

Polycyclic presentation:

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

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, 8662175);