#481284

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#576	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

949 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)(9,10,13,18,14,20,29,19)(11,21,24,35,22,16,32,34)(12,25,27,17,28,15,26,31)(23,30,40,37,38,36,39,33);(10,15)(12,26)(13,29)(16,30)(17,18)(19,31)(20,25)(21,36)(23,39)(24,32)(27,28)(33,34)(35,37)(38,40);(2,5)(6,8)(9,11)(10,16)(12,23)(13,24)(14,22)(15,30)(17,33)(18,34)(19,35)(20,21)(25,36)(26,39)(27,40)(28,38)(29,32)(31,37);(11,22)(16,21)(23,38)(24,32)(30,36)(33,37)(34,35)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216752, #72216759, #72216898, …
maximal normal subgroup of	#72216752, #72216759, #72216898, …
maximal quotient of	C₂×#481284, #72318656, #72318709, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388480);