#7897751

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	#2397	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=ghi,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, 7804947);