#8425414

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^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, 8332610);