#7732385

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

4197 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=hi,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, 7639581);