#8416823

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	#55525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324019);