#8755393

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=hi,e^2=h,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, 8662589);