#8291631

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8198827);