#6396561

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
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	#60790	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303757);