#9863172

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

3742 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9770368);