#7712349

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	#39124	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4408 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)(9,10,16,22)(11,23,24,17)(12,21,15,18)(13,19,20,14)(28,36)(31,38)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,13)(14,21)(15,20)(16,24)(18,19)(22,23)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(9,12,16,15)(10,18,22,21)(11,20,24,13)(14,23,19,17)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,13,16,20)(10,19,22,14)(11,12,24,15)(17,18,23,21);(1,3,4,7)(2,6,5,8)(9,14,16,19)(10,20,22,13)(11,21,24,18)(12,17,15,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,15,16,12)(10,21,22,18)(11,20,24,13)(14,23,19,17)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=h,e^2=g,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	#72239255
maximal normal subgroup of	#72239255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619545);