#8425628

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332824);