#483591

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#442
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216710, #72216790, #72216809, …
maximal normal subgroup of	#72216710, #72216790, #72216809, …
maximal quotient of	#72343626

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390787);