#7955027

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

4049 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7862223);