#7895236

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	#40600	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4378 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)(11,17)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26,30,31)(27,28,33,36)(29,32,38,39)(34,35,40,37);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(10,16)(15,17)(18,23)(22,24)(26,31)(27,33)(32,39)(34,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39);(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,28)(26,27)(29,37)(30,36)(31,33)(32,40)(34,39)(35,38);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,30,38)(26,32,31,39)(27,34,33,40)(28,35,36,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^2=hi,f^2=gi,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, 7802432);