#6431706

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314714, #72314796, #72314800
maximal normal subgroup of	#72314714, #72314796, #72314800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338902);