#8508904

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 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)(3,12)(4,10)(5,9)(6,8)(7,11)(13,15)(14,16)(17,18)(19,27)(20,26)(21,31)(22,24)(23,25)(28,29)(30,32)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,3,7,6)(2,8,11,12)(4,13,5,14)(9,15,10,16)(17,19,24,23)(18,25,22,27)(20,28,31,30)(21,29,26,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4,7,5)(2,9,11,10)(3,13,6,14)(8,15,12,16)(17,20,24,31)(18,26,22,21)(19,28,23,30)(25,32,27,29)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5,7,4)(2,10,11,9)(3,13,6,14)(8,15,12,16)(17,21)(18,20)(19,29)(22,31)(23,32)(24,26)(25,28)(27,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,2)(3,12)(4,9)(5,10)(6,8)(7,11)(13,16)(14,15)(17,22,24,18)(19,25,23,27)(20,21,31,26)(28,32,30,29)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,6,7,3)(2,8,11,12)(4,14,5,13)(9,15,10,16)(17,23,24,19)(18,25,22,27)(20,30,31,28)(21,29,26,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8416100);