#7712786

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4236 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)(11,19)(12,17)(13,15)(14,24)(16,20)(18,21)(22,23)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,22)(15,20)(17,24)(18,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,4,5,2)(3,7,8,6)(10,13)(11,20)(12,22)(18,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(9,12,15,22)(10,17,13,23)(11,21,20,14)(16,24,19,18);(9,13,15,10)(11,19,20,16)(12,23,22,17)(14,24,21,18)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,14,15,21)(10,18,13,24)(11,22,20,12)(16,23,19,17)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=h,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.

Relationships

maximal subgroup of	#72239901
maximal normal subgroup of	#72239901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619982);