#6449098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286090, #72286176, #72286224
maximal normal subgroup of	#72286090, #72286176, #72286224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356294);