#7927184

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	#41423	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=i,e^2=g,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, 7834380);