#8096524

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

3890 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)(3,10)(4,12)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18,23,28)(19,21,31,24)(20,26,30,25)(22,27,29,32);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(17,19,23,31)(18,24,28,21)(20,22,30,29)(25,27,26,32)(34,38)(39,45)(40,46)(47,48);(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,21,23,24)(18,19,28,31)(20,27,30,32)(22,25,29,26);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,20)(18,26)(19,22)(21,27)(23,30)(24,32)(25,28)(29,31)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,5)(2,8)(3,9)(4,13)(6,10)(7,15)(11,12)(14,16)(17,22)(18,27)(19,30)(20,31)(21,25)(23,29)(24,26)(28,32)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003720);