#481283

order	512
nilpotency class	4
isoclinism	isoclinic class of #11663
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287549, #72287557, #72287625, …
maximal normal subgroup of	#72287549, #72287557, #72287625, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388479);