#8075220

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	#44776	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982416);