#8112541

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	#52410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(21,29)(25,30)(26,31)(27,32)(34,36)(35,37)(38,46)(47,48);(2,7)(4,9)(8,10)(12,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,34)(35,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(3,11)(8,10)(12,15)(13,16)(18,23)(19,24)(25,30)(26,31)(33,35,39,37)(34,40,36,43)(38,42,46,45)(41,47,44,48);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,36,39,34)(35,40,37,43)(38,44,46,41)(42,47,45,48);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37,39,35)(34,40,36,43)(38,45,46,42)(41,47,44,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8019737);