#8096463

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

3873 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=gi,f^2=gi,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, 8003659);