#8459210

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^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, 8366406);