#9815018

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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)(17,18,23,25)(19,27,21,24)(20,30,26,22)(28,29,31,32)(33,34,38,42)(35,37,41,39)(36,40,46,48)(43,47,44,45);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,19,23,21)(18,24,25,27)(20,28,26,31)(22,29,30,32)(33,35,38,41)(34,39,42,37)(36,44,46,43)(40,47,48,45);(2,8)(3,11)(9,13)(10,12)(18,25)(20,26)(24,27)(28,31);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,20,23,26)(18,22,25,30)(19,28,21,31)(24,29,27,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,47)(46,48);(1,4)(2,9)(3,12)(5,15)(6,14)(7,16)(8,13)(10,11)(17,21,23,19)(18,24,25,27)(20,31,26,28)(22,29,30,32)(33,36,38,46)(34,40,42,48)(35,43,41,44)(37,47,39,45);(1,5,6,16)(2,10,8,12)(3,13,11,9)(4,7,14,15)(17,22)(18,26)(19,29)(20,25)(21,32)(23,30)(24,31)(27,28)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,48,44)

Polycyclic presentation:

2^9|a^2=gi,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, 9722214);