#8096649

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:

(1,2,8,10)(3,5,7,4)(6,9,14,16)(11,13,15,12)(18,23)(24,25)(26,30)(31,32);(1,3,8,7)(2,4,10,5)(6,11,14,15)(9,12,16,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,38)(39,45)(40,46)(47,48);(1,4,8,5)(2,7,10,3)(6,12,14,13)(9,15,16,11)(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,5,8,4)(2,3,10,7)(6,13,14,12)(9,11,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,14)(10,16)(17,20,22,19)(18,25,23,24)(21,27,29,28)(26,31,30,32)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,7,8,3)(2,4,10,5)(6,11,14,15)(9,13,16,12)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(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=h,c^2=g,d^2=h,e^2=gi,f^2=i,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, 8003845);