#8323533

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8230729);