#8919510

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	#51749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219287
maximal normal subgroup of	#72219283, #72219287
maximal quotient of	#72219270

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826706);