#6467252

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,15)(14,19)(18,24)(20,26)(21,29)(22,28)(23,27)(25,31)(30,34)(32,36)(33,37)(35,38)(39,40);(10,15)(11,20)(12,23)(13,24)(16,28)(17,31)(18,19)(21,35)(22,34)(25,36)(29,39)(37,38);(2,5)(6,8)(9,11,13,20,14,22,24,34)(10,16,15,26,19,28,18,30)(12,21,23,33,25,35,36,40)(17,29,27,37,31,38,32,39);(9,12,24,36,14,25,13,23)(10,17,18,32,19,31,15,27)(11,21,34,40,22,35,20,33)(16,29,30,39,28,38,26,37);(1,3,4,7)(2,6,5,8)(12,25)(17,31)(21,35)(23,36)(27,32)(29,38)(33,40)(37,39)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,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.

Relationships

maximal subgroup of	#72320520, #72320791
maximal normal subgroup of	#72320520, #72320791
maximal quotient of	#72322382, #72322424, #72322536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374448);