#7918742

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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)(11,18)(12,15)(13,20)(14,17)(16,22)(19,23)(21,24)(25,26,30,32)(27,33,35,28)(29,37,38,31)(34,36,40,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,21)(22,24);(1,3,5,4)(2,7,6,8)(10,12)(11,17)(16,20)(19,24)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,4,5,3)(2,8,6,7)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,21,24,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,13)(10,16)(11,19)(12,20)(14,23)(15,22)(17,24)(18,21)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39);(9,14,15,18)(10,17,12,11)(13,21,22,23)(16,19,20,24)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=hi,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, 7825938);