#8688197

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,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, 8595393);