#8075553

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

4089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,f^2=h,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, 7982749);