#10352527

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

3749 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:

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

Polycyclic presentation:

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

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, 10259723);