#8458882

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,e^2=ghi,f^2=i,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, 8366078);