#8932262

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	#47170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gh,e^2=gi,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, 8839458);