#6467250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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)(11,16)(12,17)(13,15)(14,19)(18,25)(20,26)(21,29)(22,28)(23,27)(24,31)(30,34)(32,36)(33,37)(35,38)(39,40);(10,15)(11,20)(12,23)(13,25)(16,28)(17,31)(18,19)(21,35)(22,34)(24,36)(29,39)(37,38);(2,5)(6,8)(9,11,13,20,14,22,25,34)(10,16,15,26,19,28,18,30)(12,21,23,33,24,35,36,40)(17,29,27,37,31,38,32,39);(9,12,13,23,14,24,25,36)(10,17,15,27,19,31,18,32)(11,21,20,33,22,35,34,40)(16,29,26,37,28,38,30,39);(1,3,4,7)(2,6,5,8)(12,24)(17,31)(21,35)(23,36)(27,32)(29,38)(33,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374446);