#7712723

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4298 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)(9,10,15,12)(11,19,20,16)(13,17,23,22)(14,21,24,18)(27,30)(29,32)(35,36)(37,39);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,12,19)(13,21,23,18)(14,22,24,17)(25,26)(27,29)(28,33)(30,32)(31,34)(35,37)(36,39)(38,40);(11,20)(13,23)(16,19)(17,22)(25,27,31,30)(26,32,34,29)(28,36,38,35)(33,37,40,39);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,19,20,16)(13,22,23,17)(14,21,24,18)(25,28,31,38)(26,33,34,40)(27,35,30,36)(29,37,32,39);(9,13,15,23)(10,17,12,22)(11,21,20,18)(14,19,24,16)(25,29,31,32)(26,27,34,30)(28,37,38,39)(33,35,40,36);(9,14,15,24)(10,18,12,21)(11,22,20,17)(13,19,23,16)(25,30,31,27)(26,32,34,29)(28,36,38,35)(33,39,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619919);