#8095272

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3775 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002468);