#8425447

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,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, 8332643);