#8504745

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411941);