#8110629

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

4654 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340988
maximal subgroup of	#72340988, #72340990
maximal normal subgroup of	#72340988, #72340990

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017825);