#8461204

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	#40164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8368400);