#8533610

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

4051 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18,23,25)(19,24,29,31)(20,26,30,32)(21,22,28,27)(35,36)(39,40)(43,44)(46,47);(2,7)(3,12)(9,13)(10,16)(17,19)(18,24)(20,21)(22,26)(23,29)(25,31)(27,32)(28,30)(33,34,38,42)(35,39,36,40)(37,41,45,48)(43,46,44,47);(2,7)(4,14)(8,11)(10,16)(18,25)(19,29)(21,28)(26,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,20)(18,26)(19,28)(21,29)(22,31)(23,30)(24,27)(25,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,21,23,28)(18,27,25,22)(19,20,29,30)(24,32,31,26)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,22,23,27)(18,28,25,21)(19,26,29,32)(20,24,30,31)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440806);