#8096623

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(5,7)(6,9)(8,14)(11,15)(13,16)(17,18,23,22)(19,26,24,20)(21,29,31,25)(27,28,30,32);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(17,19)(18,20)(21,30)(22,26)(23,24)(25,32)(27,31)(28,29)(34,38)(39,45)(40,46)(47,48);(17,20)(18,24)(19,22)(21,28)(23,26)(25,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,31)(18,25,22,29)(19,27,24,30)(20,28,26,32);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,30)(31,32)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,22,23,18)(19,26,24,20)(21,29,31,25)(27,32,30,28)(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=gh,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, 8003819);