#8445686

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	#59685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8352882);