#8505333

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3816 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=hi,e^2=h,f^2=gi,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, 8412529);