#7732094

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

4186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7639290);