#8460722

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,32,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,24,31)(18,25,22,29)(20,30,21,28)(23,27,32,26)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,24,21)(18,26,22,27)(19,30,31,28)(23,29,32,25)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,21,24,20)(18,27,22,26)(19,30,31,28)(23,29,32,25)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,30,32,28);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,23,24,32)(18,28,22,30)(19,27,31,26)(20,29,21,25)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=hi,f^2=hi,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, 8367918);