#8105930

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,16)(13,15)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,31)(28,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32)(33,36,37,44)(34,39,40,48)(35,41,42,43)(38,45,46,47);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,16,11,15)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,32,26,31);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,11,16)(17,21,22,20)(18,24,25,30)(19,27,29,28)(23,32,26,31)(34,40)(35,42)(39,48)(41,43);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013126);