#8425425

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332621);