#8159152

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	#56916	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066348);