#7917755

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=hi,e^2=h,f^2=gh,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, 7824951);