#9815017

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

3744 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)(18,23)(19,26)(25,28)(27,32)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,31,32,29)(35,41)(39,42)(43,45)(47,48);(2,8)(3,11)(9,13)(10,12)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,19)(18,24)(20,27)(21,23)(22,26)(25,29)(28,31)(30,32)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,4)(2,9)(3,12)(5,15)(6,14)(7,16)(8,13)(10,11)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,31,24,29)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,5,6,16)(2,10,8,12)(3,13,11,9)(4,7,14,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,32,28,27)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,e^2=g,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, 9722213);