#6388151

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60557	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1869 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)(9,10,14,19,15,12,24,18)(11,16,22,31,23,20,35,30)(13,17,28,34,27,25,36,33)(21,29,37,40,26,32,38,39);(2,5)(3,7)(9,11,14,22,15,23,24,35)(10,16,19,31,12,20,18,30)(13,26,28,38,27,21,36,37)(17,32,34,39,25,29,33,40);(9,12)(10,15)(11,20)(13,25)(14,18)(16,23)(17,27)(19,24)(21,32)(22,30)(26,29)(28,33)(31,35)(34,36)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,21)(12,25)(14,28)(15,27)(16,29)(18,33)(19,34)(20,32)(22,37)(23,26)(24,36)(30,39)(31,40)(35,38);(2,5)(6,8)(13,27)(17,25)(21,26)(28,36)(29,32)(33,34)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343105
maximal normal subgroup of	#72343105
maximal quotient of	#72343169

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295347);